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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 115, pp. 1--11.\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-2009/115\hfil Upper and lower solutions]
{Upper and lower solutions for a second-order three-point
singular boundary-value problem}

\author[Q. Zhang, D. Jiang, S. Weng,  H. Gao \hfil EJDE-2009/115\hfilneg]
{Qiumei Zhang, Daqing Jiang, Shiyou Weng, Haiyin Gao}

\address{Qiumei Zhang \newline
School of Science,
Changchun University, Changchun 130022, China.\newline
Department  of Mathematics,
Northeast Normal  University,
Changchun 130024,  China}
\email{zhangqm1110@yahoo.com.cn}

\address{Daqing Jiang \newline
Department  of Mathematics,
Northeast Normal University,
Changchun 130024, China}
\email{jiangdq067@nenu.edu.cn}

\address{School of Science, Changchun University,
Changchun 130022, China}
\email[Haiyin Gao]{gaohaiyinhealthy@yahoo.com.cn} 
\email[Shiyou Weng]{wengshiyou2001@yahoo.com.cn}


\thanks{Submitted January 27, 2009. Published September 12, 2009.}
\thanks{Supported by grants 10571021 from NSFC of China, and
and KLAS from Key Laboratory \hfill\break\indent
for Applied Statistics of MOE}
\subjclass[2000]{34B15, 34B16}
\keywords{Singular boundary-value problem;
 upper and lower solutions; \hfill\break\indent
 existence of solutions; superlinear}

\begin{abstract}
 We study the singular boundary-value problem
 \begin{gather*}
 u''+ q(t)g(t,u)=0,\quad  t \in (0,1),\; \eta  \in (0,1),\;\gamma >0\\
 u(0)=0,  \quad   u(1)=\gamma u(\eta)\,.
 \end{gather*}
 The singularity may appear at $ t=0$ and the function $g$ may
 be superlinear at infinity and may change sign.
 The existence of solutions is obtained via an upper and lower
 solutions method.
\end{abstract}

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

\section{Introduction}

  Motivated by the study of multi-point boundary-value problems for
linear second order ordinary differential equations,
Gupta \cite{g2} studied certain three point boundary-value problems
for nonlinear ordinary differential equations.
Since then, more general nonlinear
multi-point boundary-value problems have been studied by several
authors using the Leray-Schauder theorem, nonlinear alternative of
Leray-Schauder or coincidence degree theory. We refer the reader to
\cite{f1,f2,f3,g4,m1,m2,m3,m4}
 for some existence results of nonlinear multi-point
boundary-value problems. Recently, Ma \cite{m3} proved the existence of
positive solutions for the three point boundary-value problem
\begin{gather*}
u''+ b(t)g(u)=0,\quad    t \in (0,1)\\
u(0)=0,  \quad   u(1)=\alpha u(\eta),
\end{gather*}
where  $\eta  \in (0,1)$, $0< \alpha < 1/\eta$,
$ b \geq 0$ and $g  \geq 0$ is either  superlinear or  sublinear.
He  applied a fixed point theorem in cones.

In this paper, we study the singular three-point  boundary-value problem
\begin{equation}
\begin{gathered}
u''+ q(t)g(t,u)=0,\quad    t \in (0,1),\;\eta  \in (0,1),\;\gamma >0\\
u(0)=0,  \quad   u(1)=\gamma u(\eta).
\end{gathered}\label{e1.1}
\end{equation}
The singularity may appear at  $t=0$,  and  the function $g$
may  be superlinear at $u=\infty$ and may change sign.

Some basic results on the  singular two point boundary-value problems
were obtain in \cite{a1,j1,o1}, in all these papers the
arguments rely  on the assumption that $g(t,u)$ is positive. This
implies that the solutions are concave.   Recently, some  authors
have  studied the case when $g$ is allowed to change sign by
applying the modified upper and lower solutions method; see for
example \cite{j1}.

The present work is a direct extension of some results on the
singular two-point boundary-value problems.  As
in \cite{j1}, our technique  relies  essentially on a
 modified method of upper
and lower  solutions  method for singular three-point boundary-value
problems  which we believe is well adapted to this type of
problems.

\section{Upper and lower solutions}

Consider the three-point boundary-value problem
\begin{equation}
\begin{gathered}
u''+ f(t,u)=0,\quad    t \in (0,1),\;\eta  \in (0,1),\;\gamma \in(0,1/ \eta)\\
u(0)= A,  \quad   u(1)-\gamma u(\eta)= B.
\end{gathered}\label{e2.1}
\end{equation}

We use the following assumption:
\begin{itemize}

\item[(A1)]   $f:(0,1]\times \mathbb{R} \to  \mathbb{R}$
is a  continuous function, there exist two functions
$\alpha,\beta \in C([0,1],{\mathbb{R}})$ and
$\alpha(t)\leq\beta(t)$, for all $t\in [0,1]$, if there
exist a function  $h\in C(\,(0,1],(0, \infty) ) $,
such that
\begin{gather}
 |f(t,u)| \leq h(t)
  \quad \text{for }  \alpha(t)\leq u \leq \beta(t), \label{e2.2} \\
\lim_{t \to 0^+} t^2 h(t)=0,  \quad   \int_0^1 t h(t)dt < \infty.
\label{e2.3}
\end{gather}
\end{itemize}

We call a function $\alpha(t)$  a lower solution for
\eqref{e2.1}, if
$\alpha \in C([0,1], \mathbb{R}) \cap C^2((0,1), \mathbb{R})$, and
\begin{gather*}
\alpha''+ f(t,\alpha) \geq 0,\quad \mbox{for }  t \in (0,1),\\
\alpha(0)\leq A,  \quad   \alpha(1)-\gamma \alpha(\eta)\leq B.
\end{gather*}
Similarly,  we call a function $\beta(t)$  an upper solution for
 \eqref{e2.1}, if
 $\beta \in C([0,1], \mathbb{R}) \cap C^2((0,1), \mathbb{R})$, and
\begin{gather*}
\beta''+ f(t,\beta)\leq0,\quad \mbox{for }  t \in (0,1),\\
\beta(0)\geq A,  \quad   \beta(1)-\gamma \beta(\eta)\geq B.
\end{gather*}
A function $u(t)$ is said to be a solution to \eqref{e2.1},
if it is both a lower and an upper solution to \eqref{e2.1}.

Our first result reads as follows.

\begin{theorem} \label{thm1}
Assume  {\rm (A1)} and let
$\alpha,\beta $ be, respectively, a lower solution and an upper
solution for  \eqref{e2.1} such that $\alpha(t)\leq \beta (t)$ on
$[0,1]$.
 Then \eqref{e2.1} has at least one solution $ u(t) $
such that
$$
\alpha(t)\leq u(t)\leq \beta(t),\quad \text{for } t\in [0,1].
$$
\end{theorem}

Consider now the modified boundary-value problem
\begin{equation}
\begin{gathered}
u''+f_1(t,u)=0,\quad \text{for }  t \in (0,1),\\
u(0)=A, \quad  u(1)-\gamma u(\eta)=B,
\end{gathered} \label{e2.1*}
\end{equation}
where
$$
f_1(t,u) = \begin{cases}
f(t,\alpha(t)), & \mbox{if }  u<\alpha(t),\\
f(t,u),    & \mbox{if }  \alpha(t)\leq u \leq  \beta(t),\\
f(t,\beta(t)), & \mbox{if }  u> \beta(t).
\end{cases}
$$

\begin{lemma} \label{lem2.1}
  Assume that \eqref{e2.3} holds. Then the  boundary-value problem
\begin{equation}
\begin{gathered}
y''=-h(t), \quad   0 < t < 1,\\
y(0)=A, \quad y(1)-\gamma y(\eta)=B
\end{gathered} \label{e2.4}
\end{equation}
has a unique  solution $y(t)$ in
$C([0,1],[0, \infty) ) \cap C^2((0,1), \mathbb{R})$,
 which can be written  as
$$
y(t)= A + \frac{B-A(1-\gamma)}{1-\gamma \eta} t +\int_0^1 G(t,s)
h(s)ds, \quad  0 \leq t \leq 1,
$$
where $G(t,s)$ is Green's function of the boundary-value problem
 $- y''= 0$, $y(0)=0$, $y(1)=\gamma y(\eta)$. The function $G$
is explicitly given by:
when $0\leq s \leq \eta$,
\[
 G(t,s) = \begin{cases}
\frac{s[1-t-\gamma(\eta-t)]}{1-\gamma \eta}, &   s\leq t,\\
\frac{t[1-s-\gamma(\eta-s)]}{1-\gamma \eta}, &  s>t;
\end{cases}
\]
when $\eta<s \leq 1$,
$$
G(t,s) = \begin{cases}
\frac{s(1-t)+\gamma \eta (t-s)}{1-\gamma \eta}, &  s\leq t,\\
\frac{t(1-s)}{1-\gamma \eta}, &        s>t.
\end{cases}
$$
\end{lemma}

\begin{proof}
  Uniqueness. The proof of the uniqueness of a solution
is standard and hence omitted.
Existence. Let
$$
y(t) :=A + \frac{B-A(1-\gamma)}{1-\gamma \eta} t + \int^1_0 G(t,s)
h(s) ds, \quad  0 \leq t \leq 1;
$$
i.e.,
$$
y(t)=\begin{cases}
A + \frac{B-A(1-\gamma)}{1-\gamma \eta} t
+\int^t_0\frac{s[1-t-\gamma(\eta-t)]}{1-\gamma \eta}h(s)ds\\
+\int^\eta_t\frac{t[1-s-\gamma(\eta-s)]}{1-\gamma \eta}h(s)ds
+\int^1_\eta\frac{t(1-s)}{1-\gamma \eta}h(s)ds, & 0\leq t \leq \eta,
\\[4pt]
A + \frac{B-A(1-\gamma)}{1-\gamma \eta} t
+\int^\eta_0 \frac{s[1-t-\gamma(\eta-t)]}{1-\gamma \eta}h(s)ds\\
+\int^t_\eta \frac{s(1-t)+\gamma \eta (t-s)}{1-\gamma \eta}h(s)ds
+\int^1_t\frac{t(1-s)}{1-\gamma \eta}h(s)ds, &  \eta<t\leq 1.
\end{cases}
$$
Then we have
$$
y'(t)=\begin{cases}
\frac{B-A(1-\gamma)}{1-\gamma \eta}
+\int^t_0\frac{s(\gamma -1)}{1-\gamma \eta}h(s)ds\\
+\int^\eta_t\frac{1-s-\gamma (\eta -s)}{1-\gamma \eta}h(s)ds\\
+\int^1_\eta\frac{1-s}{1-\gamma \eta}h(s)ds, &  0< t \leq \eta,
\\[4pt]
 \frac{B-A(1-\gamma)}{1-\gamma \eta}  +\int^\eta_0
\frac{s(\gamma -1)}{1-\gamma \eta}h(s)ds \\
+\int^t_\eta \frac{\gamma
\eta -s}{1-\gamma \eta}h(s)ds +\int^1_t\frac{1-s}{1-\gamma
\eta}h(s)ds, &  \eta < t \leq 1.
\end{cases}
 $$
and $y''(t) = -  h(t)$ for all $ t \in (0,1)$.
Since  $\int_0^1 th(t)dt < \infty$,
 $ \lim_{t \to0^+}  \int_0^t s h(s) ds = 0$; so we have
$$
 y(0) = A+\lim_{t \to0^+} t
 \int^\eta_t\frac{1-s-\gamma(\eta-s)}{1-\gamma \eta}h(s)ds.
$$
 If $ \int^1_0\frac{1-s-\gamma(\eta-s)}{1-\gamma \eta}h(s)ds
 < \infty$, then $y(0) =A$.
If $ \int^1_0\frac{1-s-\gamma(\eta-s)}{1-\gamma \eta}h(s)ds
 = \infty$, then by \eqref{e2.3} we obtain
 $$
 y(0) = A+\lim_{t\to 0^+}  \frac{\int^\eta_t\frac{1-s-\gamma(\eta-s)}{1-\gamma \eta}h(s)ds} {1/t} =
  A+\lim_{t \downarrow 0^+} t^2 h(t) \frac{1-\gamma \eta+t(\gamma -1)}{1-\gamma \eta}= A.
  $$
We  have also
\begin{align*}
&y(1) -\gamma y(\eta) \\
&=\frac{B-A(1-\gamma)}{1-\gamma \eta}
 +\int^\eta_0\frac{s\gamma(1-\eta)}{1-\gamma \eta}h(s)ds
 +\int^1_\eta \frac{\gamma \eta(1-s)}{1-\gamma \eta}h(s)ds\\
&\quad -\gamma( \frac{B-A(1-\gamma)}{1-\gamma
\eta}\eta+\int^\eta_0\frac{s(1-\eta)}{1-\gamma \eta}h(s)ds
+\int^1_\eta \frac{\eta(1-s)}{1-\gamma \eta}h(s)ds)=B.
\end{align*}
 This shows that $y(t)$ is a positive solution of \eqref{e2.4},
and  $y \in  C([0,1], [0, \infty)) \cap C^2((0,1), \mathbb{R})  $.
\end{proof}

Let us define an operator $ \Phi :X \to X $ by
\begin{equation}
(\Phi u)(t) = A + \frac{B-A(1-\gamma)}{1-\gamma \eta} t + \int^1_0
G(t,s) f_1(s,u(s)) ds, \label{e2.5}
\end{equation}
where $ X = \{ u \in C([0,1],\mathbb{R})$ with the norm $\|u\| \} $
is a  Banach space, with
$$
\|u\|:= \sup\{|u(t)|:0\leq t \leq 1\}.
$$
Without loss of generality,  we assume that $A=B=0$.


To prove the existence of a solution to \eqref{e2.1*}, we
need the following Lemma.

\begin{lemma} \label{lem2.2}
The function $\Phi$ is continuous  from $X$ to $X$ and $\Phi(X)$ is a
compact subset of $X$.
\end{lemma}

\begin{proof}
 As in the proof of Lemma \ref{lem2.1}, from  the definition of $f_1$ and
 from \eqref{e2.5}, we have
\begin{equation}
|(\Phi u)(t)|  \leq \int_0^1G(t,s)|f_1(s,u(s))|ds\leq
\int_0^1G(t,s)h(s)ds = y(t), \quad t\in [0,1]. \label{e2.6}
\end{equation}
 So we have $\Phi u \in   C([0,1], \mathbb{R}) \cap C^2((0,1), \mathbb{R}),
 $ and
\begin{equation}
\|\Phi u\| \leq \|y\|. \label{e2.7}
\end{equation}
This shows that $\Phi(X)$ is a bounded subset of $X$.

Noting the facts that $y(0)=0$ and the continuity of $y(t)$ on
$[0,1]$, we have from \eqref{e2.6} that for any $\epsilon >0$,
one can find a $\delta_1>0$ (independent with $u$)
such that $0<\delta_1<1/8$ and
\begin{equation}
(\Phi u)(t)<\frac{\epsilon}{2},  \quad   t \in [0, 2 \delta_1] .
\label{e2.8}
\end{equation}
On the other hand,  from \eqref{e2.5}, since $|f_1(s,u(s))|\leq h(s)$,
 $s\in (0,1)$,  we can obtain
$$
|(\Phi u)'(t)| \leq L, \quad   t\in [\delta_1, 1].
$$
Let $\delta_2=\frac {\epsilon}{2 L} $,  then for
$t_1,  t_2 \in [\delta_1,1]$,  $|t_2-t_1|<\delta_2$,  we have
\begin{equation}
|(\Phi u)(t_1)-(\Phi u)(t_2)|\leq L |t_1- t_2|< \frac {\epsilon}{2}.
\label{e2.9}
\end{equation}
 Define $\delta =\min\{\delta_1,\delta_2\}$, then  using
\eqref{e2.8}, \eqref{e2.9}, we obtain
\begin{equation}
|(\Phi u)(t_1)-(\Phi u)(t_2)|<\epsilon, \label{e2.10}
\end{equation}
for $t_1, t_2\in [0,1]$,  $|t_1- t_2|<\delta$.
This shows that $\{(\Phi u)(t): u\in X \}$ is equicontinuous on $[0,1]$.

We can obtain the continuity of $\Phi$ in a similar way as above.
  In fact, if $u_n,  u\in X$ and
$\|u_n-u\|\to 0\ $ as $n\to \infty$, then we have
\begin{equation}
|(\Phi u_n)(t) - (\Phi u)(t)| \leq 2
 \int_0^1G(t,s)h(s)ds =2 y(t),\quad  t\in [0,1], \label{e2.11}
\end{equation}
Noting the facts that $y(0)=0$ and the continuity of $y(t)$ on
$[0,1]$, then for any $\epsilon >0$, one can find a $\delta_1>0$
(independent of $u_n $)
 such that $0<\delta_1<1/ 8$ and
\begin{equation}
|(\Phi u_n)(t) - (\Phi u)(t)|
  < \epsilon,   \quad  t \in [0,  \delta_1]. \label{e2.12}
\end{equation}
On the other hand, from  the continuity of $f_1$, one has
\begin{equation}
|(\Phi u_n)(t) -( \Phi u)(t)|   \to  0, \quad    t \in [\delta_1,
1], \label{e2.13}
\end{equation}
as $n\to \infty$. This together with \eqref{e2.12}
 implies that $\|\Phi u_n-\Phi u\| \to 0$
as $n\to \infty$.
Therefore, $\Phi :X\to X$ is completely continuous.
  The proof   is complete.
\end{proof}

\begin{lemma} \label{lem2.3}
 Let $u(t)$ be a solution to \eqref{e2.1*}. Then
$\alpha(t) \leq u(t)\leq \beta(t)$ for all $t\in [0,1]$;
 i.e., $u(t)$ is a solution to \eqref{e2.1}.
\end{lemma}

\begin{proof}
We first prove that $u(t)\leq \beta(t)$ on $[0,1]$. Let
$x(t):=u(t)-\beta(t)$. Assume that $u(t)>\beta(t)$ for some
$t\in [0,1]$. Since  $u(0)=0\leq  \beta (0)$, it follows that
 $$
x(0)\leq 0, \quad
x(1)=u(1)-\beta(1)\leq \gamma u(\eta)-\gamma \beta(\eta)=\gamma
x(\eta).
$$
Let $ \sigma  \in (0,1]$ be such that
$ x(\sigma)=\max_{t \in [0,1]}x(t)$. Then $x(\sigma) >0$.

   Case(i):  $ \sigma  \in (0,1)$. So there exists an interval
$(a,\sigma]\subset (0,1)$ such that $x(t)>0$ in $(a,\sigma]$, and
$$
x(a)=0, \quad   x(\sigma)=\max_{t \in [0,1]} x(t) >0, \quad
  x'(\sigma)=0.
$$
For $t\in (a, \sigma]$ we have that $f_1(t,u(t))=f(t,\beta(t))$ and
therefore
$$
u''(t)+f_1(t,u(t))=u''(t)+f(t,\beta(t))=0 \quad\text{for all  }
 t\in (a, \sigma].
$$
On the other hand, as $\beta$ is an upper solution for \eqref{e2.1}, we
 have
$$
\beta''(t)+f(t,\beta(t))\leq 0   \quad\text{for all }
t\in (a, \sigma].
$$
Thus, we obtain
$u''(t)\geq \beta''(t)$  for   all   $t\in (a, \sigma]$,
and hence, $x''(t)\geq0$. Then $x'(t)\leq 0$ on $(a,1] $
which is a contradiction.

    Case(ii):  $ \sigma =1$. So there exists $(a,1] \subset (0,1]$
such that
$$
x(a)=0, \quad x(1)=\max_{t \in [0,1]} x(t), x(1)- \gamma
x(\eta) \leq 0.
$$
In the same way as in Case(i), we can obtain that
$ x(t) >0,x''(t)\geq 0,  t \in (a,1]$. Since
$ x(\eta)\geq \frac{1}{\gamma}x(1)>0$, then $ \eta >a$.
\end{proof}

  Consider the three-point boundary-value problem
\begin{equation}
\begin{gathered}
x''=h(t)>0, \quad   a < t < 1, \\
x(a)=0,  \quad  x(1)-\gamma x(\eta)=b_1 \leq 0.
\end{gathered} \label{e2.14}
\end{equation}
Then this equation has a unique  solution
$x(t) \in  C([a,\sigma], [0, \infty)) \cap
C^2((a,1), \mathbb{R})$,  which can be represented as
$$
x(t)= \frac{b_1 (t-a)}{1-a- \gamma (\eta-a)}  -\int_a^1
G_{[a,1]}(t,s) h(s)ds, \quad  a \leq t \leq 1,
$$
where $G_{ [a,1]}(t,s)$ is the Green's function of the
boundary-value problem
$- y''= 0$, $y(a)=0$, $y(1)=\gamma y(\eta)$,
which is explicitly given by:
when $a\leq s \leq \eta$,
$$
G_{ [a,1]}(t,s) = \begin{cases}
\frac{(s-a)[1-t-\gamma(\eta-t)]}{1-a- \gamma (\eta-a)}, &   s\leq t,\\[4pt]
\frac{(t-a)[1-s-\gamma(\eta-s)]}{1-a- \gamma (\eta-a)}, &   s>t;
\end{cases}
$$
when $\eta<s \leq 1$,
$$
G_{ [a,1]}(t,s) = \begin{cases}
\frac{(s-a)(1-t)+\gamma(t-s)(\eta-a)}{1-a- \gamma (\eta-a)}, & s\leq t,\\[4pt]
\frac{(t-a)(1-s)}{1-a- \gamma (\eta-a)};               s>t.
\end{cases}
$$
Since $0< \gamma < \frac{1}{\eta}< \frac{1-a}{\eta-a}$,
then $G_{ [a,1]}(t,s) \geq 0$, and hence
 $ x(t)\leq 0 $ on $[a,1]$, which is a contradiction.
In very much the same way, we can prove that
 $u(t) \geq \alpha(t)$ on $[0,1]$.

\section{Main results}

Let $g:[0,1] \times (0, \infty)\to { \mathbb{R}}$  be a continuous
function and $q \in C((0,1], { \mathbb{R}_0^+})$.
Consider the three-point boundary-value problem
\begin{equation}
\begin{gathered}
u''+ q(t)g(t,u)=0,\quad    t \in (0,1),\;\eta  \in (0,1),\;
\gamma \in(0,1]\\
u(0)=0, \quad  u(1)=\gamma u(\eta).
\end{gathered}\label{e3.1}
\end{equation}

\begin{theorem} \label{thm2}
 Assume that
\begin{itemize}

\item[(H1)]   $|g(t,x)| \leq   F(x) + Q(x)$ on
$ [0,1] \times (0,\infty)$ with $F>0$ continuous and non-increasing
on $(0, \infty)$, $Q \geq 0$ continuous on $[0, \infty)$, and
$\frac{Q}{F}$ nondecreasing on $(0,\infty)$;


\item[(H2)]  there exist  constants $L > 0$ and $\varepsilon>0$ such
that
$g(t,x) > L$ for all  $(t,x)  \in [0,1] \times (0,\varepsilon]$,
and $F(x)>L$,  $x\in (0,\varepsilon]$;

\item[(H3)]
\begin{gather}
\lim_{t\to 0^+} t^2q(t)=0,  \quad   \int_0^1 tq(t)dt<\infty,\label{e3.2}\\
\sup_{c \in (0, \infty)} \Big( \frac{1}
 {1 + \frac{Q(c)} {F(c)} }
\int^c_0 \frac{du}{ F(u)} \Big) > b_0, \label{e3.3}
\end{gather}
where
$b_0 = \int^1_0 rq(r)dr$.
\end{itemize}
Then  \eqref{e3.1} has at least one  solution
$u \in C([0,1], [0,\infty)) \cap C^2((0,1),  \mathbb{R})$
with $u(t)> 0$ on $(0,1]$.
\end{theorem}

 From Lemma \ref{lem2.1}, we obtain the following result.

\begin{lemma} \label{lem3.1}
 There exists an unique  solution
$W\in C([0,1],[0,\infty))\cap C^2((0,1),\mathbb{R})$,
with $W(t)> 0$ on $(0,1]$ to the problem
\begin{equation}
\begin{gathered}
W''+ q(t)=0, \quad   0 < t < 1,\\
W(0)=0, \quad  W(1)=\gamma W(\eta).
\end{gathered} \label{e3.4}
\end{equation}
\end{lemma}

Choose $M>0$, $\delta >0$ ($\delta <M$) such that
\begin{equation}
\frac{1}{1+\frac{Q(M)}{F(M)}} \int_{\delta}^{M}\frac{du}{F(u)}> b_0.
\label{e3.5}
\end{equation}
Let $n_0\in \{1,2,\dots \}$ be chosen so that
$1/n_0 < \min\{\varepsilon-m\|W\|, \delta\}$,
where $W$ is the solution of \eqref{e3.4}, and
$0<m< \min\{ L,\varepsilon/\|W\|, 1 \}$ is chosen and
fixed. Let $N^+=\{n_0,n_0+1,\dots \}$.


We first show that the  boundary-value problem
\begin{equation}
\begin{gathered}
u''+ q(t)g(t, u)=0, \quad   0 < t < 1,\\
u(0)=\frac{1}{n}, \quad  u(1)-\gamma u(\eta)=\frac{1-\gamma }{n}, \quad
 n\in N^+
\end{gathered} \label{e3.6n}
\end{equation}
has a solution $u_n$ for each $n\in N^+$ with
 $u_n(t)\geq \frac{1}{n}$ for $t\in [0,1]$ and $\|u_n\|<M$.

We have the following Claim

\noindent\textbf{Claim:}
Let $\alpha_n(t)=mW(t)+\frac{1}{n}$,  $t\in [0,1]$,
then $\alpha_n(t)$ is a (strict) lower solution for problem
\eqref{e3.6n}.

\begin{proof}
For the choice of $m$ and $n$, we have
$mW(t)+\frac{1}{n}\leq m\|W\|+\frac{1}{n_0}<\varepsilon$,
then from (H2),
$$
g(t, mW(t)+\frac{1}{n}) > L > m  \quad\text{for all }  t\in [0,1].
$$
Then we obtain
\begin{align*}
\alpha_n''(t)+ q(t)g(t, \alpha_n(t))
&=(mW(t)+\frac{1}{n})''+q(t)g(t,mW(t)+\frac{1}{n})\\
&=mW''(t)+q(t)g(t,mW(t)+\frac{1}{n})\\
&=q(t)(g(t,mW(t)+\frac{1}{n})-m)>0, \quad  0<t<1.
\end{align*}
We obtain
 $\alpha_n(0)=mW(0)+\frac{1}{n}=\frac{1}{n}$, and
\begin{align*}
\alpha_n(1)-\gamma \alpha_n(\eta)
&=mW(1)+\frac{1}{n}-\gamma(mW(\eta)+\frac{1}{n})\\
&= m(W(1)-\gamma W(\eta))+\frac{1-\gamma}{n}= \frac{1-\gamma}{n}.
\end{align*}
 Thus the proof of Claim is complete.
\end{proof}

To find the upper solution of \eqref{e3.6n}, we consider
the  problem
\begin{equation}
\begin{gathered}
u''+ q(t)F(u)(1+\frac{Q(M)}{F(M)})=0, \quad  0 < t < 1,\\
u(0)=\frac{1}{n},    u(1)-\gamma u(\eta)=\frac{1-\gamma }{n}.
\end{gathered} \label{e3.7n}
\end{equation}
To show that this problem has a solution we study
\begin{equation}
\begin{gathered}
u''+ q(t)F^*(u)(1+\frac{Q(M)}{F(M)})=0, \quad  0 < t < 1,\\
u(0)=\frac{1}{n},  u(1)-\gamma u(\eta)=\frac{1-\gamma }{n},
\end{gathered} \label{e3.8n}
\end{equation}
where
$$
F^*(u)=\begin{cases}
F(u),& u\geq 1/n,\\
F(\frac{1}{n} ),& u < 1/n.
\end{cases}
$$
Then $F^*(u)\leq F(u)$ for $u>0$.

In the same way as in the Claim, we can easily prove
$\alpha_n(t)=\frac{1}{n}+mW(t)$ is also a (strict) lower solution of
\eqref{e3.8n}.

By Lemma \ref{lem2.1}, let
$\beta_n^0\in C([0,1],\mathbb{R})\cap C^2((0,1),\mathbb{R})$ be the
unique solution of the boundary-value problem
\begin{equation}
\begin{gathered}
u''+ q(t)F(\alpha_n(t))(1+\frac{Q(M)}{F(M)})=0, \quad  0 < t < 1,\\
u(0)=\frac{1}{n}, \quad u(1)-\gamma u(\eta)=\frac{1-\gamma }{n}.
\end{gathered} \label{e3.9n}
\end{equation}
Since $\beta_n^0$ is a solution of this equation,
\begin{gather*}
{\beta_n^0}''+ q(t)F(\alpha_n(t))(1+\frac{Q(M)}{F(M)})=0, \quad
   0< t <1,\\
{\beta_n^0}(0)=\frac{1}{n}, \quad  {\beta_n^0}(1)-\gamma
{\beta_n^0}(\eta)=\frac{1-\gamma}{n}.
\end{gather*}
On the other hand, as $\alpha_n$ is a lower solution of
\eqref{e3.8n},
and $\alpha_n\geq 1/n$,  we  have
\begin{gather*}
{\alpha_n}''+ q(t)F(\alpha_n(t))(1+\frac{Q(M)}{F(M)})\geq 0,  \quad  0< t<1,\\
\alpha_n(0)=\frac{1}{n}, \quad \alpha_n(1)-\gamma
{\alpha_n}(\eta)=\frac{1-\gamma}{n}.
\end{gather*}
So we obtain $\alpha_n(t)\leq \beta_n^0(t)$ for $t\in [0,1]$. Thus
\begin{align*}
&{\beta_n^0}''+q(t)F^*(\beta_n^0)(1+\frac{Q(M)}{F(M)})\\
&=-q(t)F(\alpha_n)(1+\frac{Q(M)}{F(M)})+q(t)F(\beta_n^0)
 (1+\frac{Q(M)}{F(M)})\\
&=q(t)(1+\frac{Q(M)}{F(M)})(F(\beta_n^0)-F(\alpha_n))\leq 0,
\end{align*}
so that $\beta_n^0$ is an upper solution for problem \eqref{e3.8n}.


If we now take $\alpha_n^0\equiv \alpha_n$, we have that
$\alpha_n^0$ and $\beta_n^0$ are, respectively, a lower and an upper
solution of \eqref{e3.8n} with
$\alpha_n^0(t)\leq \beta_n^0(t)$, for all
$t\in [0,1]$. So by the Lemma \ref{lem2.3}, we know that there exists a
solution $\beta_n\in C([0,1],\mathbb{R})\cap C^2((0,1),\mathbb{R})$
of \eqref{e3.8n} such that
$$
\alpha_n(t) =\alpha_n^0(t)\leq \beta_n(t)\leq
\beta_n^0(t), \quad  \forall t\in [0,1].
$$
Now we claim that $\|\beta_n\|<M$. Suppose this is false; i.e.,
suppose $\|\beta_n\|\geq M$.  Since
$\beta_n(1)-\frac{1}{n} = \gamma (\beta_n(\eta)-\frac{1}{n} )
\leq \beta_n(\eta)-\frac{1}{n},
  \beta_n''(t)\leq 0$ on (0,1) and
$\beta_n\geq \frac{1}{n}$ on $[0,1]$,
there exists $\sigma\in (0,1)$ with
$\beta_n'(t)\geq 0$ on $(0,\sigma), \beta_n'(t)\leq 0$ on
$(\sigma,1)$ and $\beta_n(\sigma)=\|\beta_n\|$.

 Then for $ z\in (0,1)$, we have
\begin{equation}
-\beta_n''(z)\leq F( \beta_n(z))(1+\frac{Q(M)}{F(M)})q(z).
\label{e3.10}
\end{equation}
Integrate from $t(0<t\leq \sigma)$ to $\sigma$  to obtain
$$
\beta_n'(t)\leq
(1+\frac{Q(M)}{F(M)})\int_t^{\sigma}F(\beta_n(z))q(z)dz;
$$
so we have
$$
\frac{\beta_n'(t)}{F(\beta_n(t))}\leq
(1+\frac{Q(M)}{F(M)})\int_t^{\sigma}q(z)dz\,.
$$
Then integrate from 0 to $\sigma$  to obtain
$$
\int_{\frac{1}{n}}^{\beta_n(\sigma)}\frac{dy}{F(y)}
\leq \big(1+\frac{Q(M)}{F(M)}\big)\int_0^{\sigma}
\Big(\int_t^{\sigma}q(z)dz\Big)dt
=\big(1+\frac{Q(M)}{F(M)}\big)\int_0^\sigma
tq(t)dt.
$$
Consequently
\begin{equation}
\int_{\delta}^{M}\frac{dy}{F(y)}\leq
\big(1+\frac{Q(M)}{F(M)}\big) \int_0^1
tq(t)dt. \label{e3.11}
\end{equation}
This  contradicts \eqref{e3.5} and consequently
$ \|\beta_n\|< M$.

  It follows from the fact $\beta_n\geq 1/n$, we can obtain $\beta_n$
is a solution of \eqref{e3.7n} also. Since $F$ is nonincreasing on
$(0,\infty)$, similar to the proof of Lemma \ref{lem2.3}, we can obtain  the
uniqueness of solutions to \eqref{e3.7n}.

Next we show that $\beta_n$ is an upper solution of \eqref{e3.6n}.
Observe that
$$
|g(t,x)|\leq F(x)+Q(x) \quad\text{on }   [0,1]\times (0,\infty).
$$
We have
\begin{align*}
\beta_n''(t) + q(t)g(t, \beta_n(t))
& \leq  - q(t)F(\beta_n(t)) \Big( 1 + \frac{Q(M)} {F(M)}\Big) +
q(t)|g(t,\beta_n(t))|\\
&\leq q(t) F( \beta_n(t)) \Big(   \frac{Q(\beta_n(t))} {F(\beta_n(t))} -
   \frac{Q(M)} {F(M)}\Big)
\leq 0,\quad      t\in (0,1).
\end{align*}
Thus $\beta_n$ is an upper solution for problem \eqref{e3.6n}.

This together with the Claim yields that  $\alpha_n$ and $\beta_n$
are, respectively, a lower and an upper solution for
\eqref{e3.6n} with $\alpha_n\leq \beta_n$ for all $t\in [0,1]$. So we
conclude \eqref{e3.6n} has a solution
$u_n\in C([0,1],\mathbb{R})\cap C^2((0,1),\mathbb{R})$ such that
$$
mW(t)+\frac{1}{n} = \alpha_{n}(t) \leq u_{n}(t) \leq
\beta_{n}(t)\leq M ,  \forall  t \in [0,1].
$$
Consider now the pointwise limit
\begin{equation}
z (t):= \lim_{n \to+ \infty} u_n(t), \quad  \forall t \in [0,1].
\label{e3.12}
\end{equation}
Let $e = [a,1] \subset (0,1]$,
Let $t_n \in (a,1) $ such that
$u'_n(t_n) = \big(u_n(1) -u_n(a)\big)/(1-a)$.
We obtain
$$
u'_n(t) = \frac{u_n(1) -u_n(a)}{1-a} + \int_t^{t_n} q(s)g(s,u_n(s)
ds, \quad  t\in e.
$$
Since $mW(t) \leq u_n(t) \leq M$,  then we have
\begin{equation}
|u'_n(t)|  \leq  \frac{2M}{1-a} + \Big(1 +  \frac{Q(M)} {F(M)}
\Big) \int_a^{1}  q(t)  F(m W(t))dt:=C(a,1)
, \quad  t\in e. \label{e3.13}
\end{equation}
Set $v_n = \max_{t\in e} |u'_n(t)|$,  which implies $v_n $ is
bounded. That means $u_n'(t)$ is bounded on $e$.

Then, by the Ascoli-Arzela theorem, it is standard to conclude that
$z(t)$  is a solution of
\eqref{e3.1}  on the interval $e = [a,1]$. Since $e$ is arbitrary,
 we find that
\[
z\in  C((0,1], [0,\infty))  \cap C^2((0,1), \mathbb{R}),  \quad
\text{and}\quad
z''(t) + q(t)g(t, z(t)) = 0, \quad   t \in (0,1).
\]
Also, we have
$$
z(0)  = \lim_{n \to + \infty} \frac{1}{n} = 0, \quad
   z(1)-\gamma z(\eta )=\lim_{n \to + \infty}
\frac{1-\gamma}{n} = 0\,.
$$
The same argument as in \cite{j1} works, we can prove the continuity of
$z(t)$ at $t=0$ and $t=1$.
The proof  is complete.

  By essentially the same argument as in Theorem \ref{thm2} and
\cite[Theorem 4.2]{a2}, we have the following result.

\begin{theorem} \label{thm3}
Assume that
\begin{itemize}
\item[(H1*)]  for any $r > 0$ there is $h_r \in C((0,1], (0,\infty) )$:
$|q(t)g(t,x)| \leq h_r(t)$ for all
$(t,x)\in (0,1]\times[r,\infty)$,
such that
$$
\lim_{t\to 0^+}t^{2}h_r(t)=0 , \quad
\int_0^1 t h_r(t) dt < +\infty;
$$

\item[(H2*)] there exist  constants $L > 0$ and $\varepsilon>0$ such
that
$g(t,x) > L$ for all   $(t,x)  \in [0,1] \times (0,\varepsilon]$.

\end{itemize}
Then  \eqref{e3.1} has at least one  solution
$u \in C([0,1], [0,\infty) \cap C^2((0,1), \mathbb{R})$.
Moreover, if $g(t,x)$ is
non-increasing in $x > 0$, then the solution is unique.
\end{theorem}

\section{An example}

Consider the  singular boundary-value problem
\begin{equation}
\begin{gathered}
u''+ \sigma t^{-m}  (u^{-\alpha}+u^{\beta} -T\sin(8\pi t))=0,
\quad  t \in (0,1)\\
u(0)=0,\quad z u(1)=\gamma u(\eta), \quad \eta  \in (0,1),\;
\gamma \in(0,1]
\end{gathered} \label{e4.1}
\end{equation}
with  $ 0 \leq m <2$,  $ \sigma > 0$,  $\alpha>0$,   $\beta \geq 0$.
Set
\begin{gather*}
  F(u) =  u^{-\alpha}, \quad  Q(u)=   u^{\beta}+1, \quad
  q(t)= \sigma t^{-m},\\
b_0 = \int^1_0 r q(r)dr =\frac{\sigma}{2-m}.
\end{gather*}
Applying  Theorem \ref{thm2},
  we  find that  \eqref{e4.1} has a
 positive solutions if
\begin{equation}
\sigma < (2-m)  \sup_{x\in (0, \infty)}
\frac{ x^{\alpha + 1}} { (\alpha+1)(1 + x^{\alpha}
 + x^{\alpha + \beta})}.
\label{e4.2}
\end{equation}
Obviously, (H1)-(H3) in  Theorem \ref{thm2} are satisfied. Thus,
\eqref{e4.1} has a solution
$u\in C([0,1],[0,\infty)\cap C^2((0,1),\mathbb{R})$
with $u>0$ on $(0,1]$.

We remark that if $ 0 \leq \beta <1$, then \eqref{e4.1} has at least one
positive solution for all $ \sigma >0$, since the right-hand side
of \eqref{e4.2} is infinity.

\begin{thebibliography}{00}

 \bibitem{a1} R. P. Agarwal and D. O'Regan:  Nonlinear
superlinear singular and nonsingular second order boundary-value problems.
 {\it J. Differential  Equations},  \textbf{143} (1998),
60-95.

\bibitem{a2} R. P. Agarwal, H. Gao, D. Q. Jiang, D. O'Regan,
 and X.  Zhang:
Existence principles and  a upper and lower solution
 theory for the one-dimension $p$-Laplacian
singular boundary-value problem with sign changing nonlinearities.
 {\it Dynamic Systems and Applications},  \textbf{15} (2006), 3-20.

\bibitem{f1}  W. Feng  and  J. R. L. Webb:
Solvability of a three-point Nonlinear boundary-value problems
 at resonance. {\it Nonlinear Analysis TMA}, \textbf{30} (1997)
No. 6, 3227-3238.

\bibitem{f2} W. Feng and J. R. L. Webb:
 Solvability of a m-point boundary-value problems
with nonlinear growth. {\it J. Math. Anal. Appl.},
 \textbf{212} (1997), 467-480.

 \bibitem{f3}  W. Feng:  On a m-point Nonlinear boundary-value problem.
{\it Nonlinear Analysis TMA}, \textbf{30} (1997), No. 6, 5369-5374.

\bibitem{g1} Y. P. Guo, W. R. Shan, W. G. Ge:
Positive solutions for a second order m-point boundary-value problems.
{\it J. comput. Appl. Math. }, \textbf{151} (2003), 415-424.

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

 \bibitem{g3} C. P. Gupta, S. K. Ntouyas, P. C. Tsamatos:
 Solvability of a m-point boundary-value problem for second order ordinary differential equations. {\it J.
Math. Anal. Appl.}, \textbf{189} (1995), 575-584.

 \bibitem{g4} C. P. Gupta:
A sharper condition for the solvability of a three-point
second-order boundary-value problem. {\it J. Math. Anal. Appl.},
\textbf{205} (1997), 586-597.

\bibitem{h1} X. M. He, W. G. Ge:
Triple solutions for second order three-point boundary-value problems.
{\it J. Math. Anal. Appl.}, \textbf{268} (2002), 256-265.

\bibitem{j1} D. Q. Jiang:  Upper and lower solutions for a
 superlinear singular  boundary-value problem.
 {\it Computers and Mathematics with Applications}, \textbf{41} (2001),
563-569.

 \bibitem{m1} R. Ma:
Existence theorems for a  second order three-point boundary-value problem.
{\it J. Math. Anal. Appl.}, \textbf{212} (1997), 430-442.

\bibitem{m2} R. Ma:
Existence theorems for a  second order m-point boundary-value problem.
{\it J. Math. Anal. Appl.}, \textbf{211} (1997), 545-555.

 \bibitem{m3} R. Ma:  Positive solutions of a nonlinear
three-point boundary-value problem.
{\it Electronic Journal of Differential Equations},
 \textbf{1999} (1999), No. 34, 1-8

 \bibitem{m4} R. Ma and N. Cataneda:
Existence of solution for nonlinear m-point boundary-value problem.
{\it J. Math. Anal. Appl.}, \textbf{256} (2001), 556-567.

 \bibitem{m5} R. Ma and Wang Haiyan:
Positive solutions of nonlinear three-point boundary-value problems.
{\it J. Math. Anal. Appl. }, \textbf{279} (2003), 1216-1227.

 \bibitem{o1} D. O'Regan:  Positive  solutions to singular and
nonsingular second-order boundary-value problems.  {\it J. Math.
Anal. Appl.},  \textbf{142} (1989), 40-52.

\bibitem{x1}  Xu Xian:  Positive solutions of Singular m-point
Boundary Value Problems with Positive Parameter.  {\it J. Math.
Anal. Appl. }, \textbf{291} (2004), 352-367.

\bibitem{z1}  Z. X. Zhang  and J. Y. Wang:
The upper and lower solution method for a
 class of singular nonlinear second-order three-point
 boundary-value problems.
   {\it J. Comput. Appl.}, \textbf{147} (2002), 41-52.

\bibitem{w1} Z. L. Wei, Z. T. Zhang:
A necessary and sufficient condition for the existence of positive
solutions of singular superlinear boundary value problems.
{\it Acta Mathematica Sinica Chinese Series},
\textbf{48} (2005), No. 1, 25-34.

\end{thebibliography}

\end{document}