\documentclass[reqno]{amsart}
\usepackage{graphicx}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 156, 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-2007/156\hfil Positive solutions]
{Positive solutions for singular three-point boundary-value problems}

\author[Z. Zhao \hfil EJDE-2007/156\hfilneg]
{Zengqin Zhao}

\address{Zengqin Zhao \newline
School of Mathematical Sciences, Qufu Normal University\\
Qufu, Shandong, 273165, China} 
\email{zqzhao@mail.qfnu.edu.cn}

\thanks{Submitted February 5, 2007. Published November 21, 2007.}
\thanks{Supported by grants: 10471075 from the National Natural Science
Foundation of China, \hfill\break\indent
Y2006A04 from the the Natural Science Foundation
of Shandong Province of China, \hfill\break\indent
20060446001 from the Doctoral Program
Foundation of Education Ministry of China}

\subjclass[2000]{34A05, 34B27, 34B05, 34B15}
\keywords{Green's function; second-order three-point boundary value problem;
\hfill\break\indent
explicit solution; iteration}

\begin{abstract}
 In this paper, we present the Green's functions for a second-order
 linear differential equation with three-point boundary conditions.
 We give exact expressions of the solutions for the linear three-point
 boundary problems by the Green's functions. As applications,
 we study uniqueness and iteration of the solutions for a nonlinear
 singular second-order three-point boundary value problem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

The Green's function plays an important role in solving
boundary-value problems of differential equations. The exact
expressions of the solutions for some linear ordinary differential
equations boundary value problems can be denoted by Green's
functions of the problems (see \cite{wingsum,BingCMA,Yang}). The
Green's function method might be used to obtain an initial
estimate for shooting method. the Green's function method for
solving the boundary value problem is an effective tool in
numerical experiments \cite{Lee}. Some boundary value problems for
nonlinear differential equations can be transformed into the
nonlinear integral equations the kernel of which are the Green's
functions of corresponding linear differential equations. The
integral equations can be solved by to investigate the property of
the Green's functions (see \cite{bai,guo88,bong,kaufmann,ruyun}).
The concept, the significance and the development of Green's
functions can be seen in \cite{Pokornyi}. The other study of
second-order three-point boundary value problems can  be seen in
\cite{Yanping04,Webb,Bing06,singh,xan05,XianSun04} and its
references. In above literatures, the three-point boundary values
are all same conditions $u(0)=0, u(1)=k u(\eta)$, the
investigation on the boundary condition $u'(0)=0$,
$u(1)=k u(\eta)$ can be seen \cite{bai1,bing1,bing2,yps}, the
investigation for other three-point boundary conditions is few,
since people may be not familiar with their Green's functions. The
solutions of the Green's functions diffuse in the literature,
there is a lack of uniform method. The undetermined parametric
method we use in this paper is a universal method, the Green's
functions of many boundary value problems for ordinary
differential equations can obtained by similar the method. In
addition, our Green's functions have orderly expressions.



We consider the Green's function for the following second-order
linear differential equation three-point boundary value problems
\begin{equation}\label{E}
u''+f(t)=0,\quad t\in [a,b],
\end{equation}
subject to the boundary value conditions, respectively,
\begin{gather}\label{B11}
u(a)=0,\quad u'(b)=k u(\eta); \\
\label{B12}
u(a)=k  u(\eta),\quad u'(b)=0; \\
\label{B13}
u(a)=0,\quad u'(b)=k u'(\eta); \\
\label{B14}
u(a)=k u'(\eta),\quad u'(b)=0; \\
\label{B21}
u'(a)=0,\quad u(b)=k  u(\eta); \\
\label{B22}
u'(a)=k  u(\eta),\quad u(b)=0; \\
\label{B23}
u'(a)=0,\quad u(b)=k  u'(\eta);\\
\label{B24}
u'(a)=k u'(\eta),\quad u(b)=0;
\end{gather}
where $a<\eta<b$ and $k$ is a constant.


This paper is organized as follows. In \S 2, we study the Green's
function for the equation  \eqref{E} satisfying the three-point
boundary conditions \eqref{B11} and give the expression of the
unique solution by the Green's function, that incarnate the
general method of deriving the Green's functions for a class of
boundary problems. In \S 3, for some interrelated boundary
conditions, we give the Green's functions of the problems
directly, omitting the particular of derivation. The correctness
of the Green's functions only need direct verification. As
applications, in \S 4, we study  the uniqueness of the solutions,
the iteration and the rate of convergence by the iteration for a
nonlinear singular second-order three-point boundary value
problem.


\section{The Green's Function of Equations \eqref{E} with the
Boundary Condition \eqref{B11}}

About the boundary value problem \eqref{E}-\eqref{B11}, we have
the following conclusions.

\begin{theorem}\label{thm1}
If $k(\eta-a)\neq 1$, then the second-order linear three-point
boundary value problem \eqref{E}\eqref{B11} has a unique solution
$u(t)$, which is given via
\begin{equation}\label{u}
u(t)=\int^1_0G_1(t,s)f(s)ds,
\end{equation}
where
\begin{gather}\label{G1}
G_1(t,s)=K(t,s)+\frac{k (t-a)}{1-k (\eta-a)}K(\eta,s),\\
\label{K}
K(t,s)=\begin{cases}  s-a, & a\le s\le  t\le b \\
 t-a, & a\le t<s\le b.
 \end{cases}
\end{gather}
\end{theorem}

\begin{remark} \label{rmk1} \rm
We call $G_1(t,s)$ the Green's function of the boundary value problem
\eqref{E}-\eqref{B11}.
\end{remark}



\begin{proof}
It is well known that the second-order two-point linear boundary value
problem
\begin{gather*}
 u''+f(t)=0,\quad t\in [a,b],\\
 u(a)=0,\quad u'(b)=0
\end{gather*}
has a unique solution
\begin{equation}\label{w}
w(t)=\int^b_aK(t,s)f(s)ds.
\end{equation}
where $K(t,s)$ is as described in (\ref{K}).
 From (\ref{w}) we obtain
\begin{equation}\label{wa}
w(a)=0,\quad w'(b)=0,\quad w(\eta)=\int^b_aK(\eta,s)f(s)ds.
\end{equation}

Assume that $u(t)$ is a solution of problem \eqref{E}-\eqref{B11}.
Then $u''(t)=w''(t)=-f(t)$, thus, we can be assume that
\begin{equation}\label{ass}
u(t)=w(t)+c+d t,
\end{equation}
where $c, d$ are constants that will be determined. From (\ref{ass})
we know that
\begin{equation}\label{u'}
u'(t)=w'(t)+d.
\end{equation}
Equations (\ref{wa}), (\ref{ass}) and (\ref{u'}) imply
\begin{gather*}
u(a)=c+d a,\\
u'(b)=d,\\
u(\eta)=c+d \eta+w(\eta).
\end{gather*}
Putting these into \eqref{B11} yields
\begin{gather*}
 c+d a=0,\\
 d=k (c+d \eta+w(\eta)).
\end{gather*}
Since $k(\eta-a)\neq 1$,
solving the system of linear equations on the unknown numbers $c, d$,
we obtain
\begin{gather*}
 c=\frac{-a k w(\eta)}{1-k (\eta-a)},\\
 d=\frac{k w(\eta)}{1-k (\eta-a)},
\end{gather*}
hence, $c+d t=\frac{k (t-a)}{1-k (\eta-a)}w(\eta)$.
Putting this into (\ref{ass}), we obtain
$$
u(t)=w(t)+\frac{k (t-a)}{1-k (\eta-a)}w(\eta),
$$
which a solution of \eqref{E}-\eqref{B11}.
This together with (\ref{w}) imply
\begin{equation}\label{iu}
u(t)=\int^b_aK(t,s)f(s)ds+\frac{k (t-a)}{1-k (\eta-a)}
\int^b_aK(\eta,s)f(s)ds.
\end{equation}
Consequently, (\ref{u}) holds.

The uniqueness of a solutions \eqref{E} \eqref{B11}
follows from the fact that the corresponding
homogeneous problem has only the trivial solution.
\end{proof}

 From Theorem \ref{thm1} we obtain the following corollary.

\begin{corollary}\label{coro2}
Suppose the nonlinear function $g(t,u)$ is continuous on $[a,b]\times R$,
then if $k (\eta-a)\neq 1$, the nonlinear three-point boundary-value problem
\begin{gather*}
u''+g(t,u)=0,\quad t\in [a,b],\\
u(a)=0,\quad u'(b)=k u(\eta)
\end{gather*}
is equivalent to the nonlinear integral equation
$$
u(t)=\int^b_aG_1(t,s) g(s,u(s))ds
$$
with $G_1(t,s)$ as in (\ref{G1}).
\end{corollary}


\begin{example} \label{exa1} \rm
 The second-order three-point linear boundary value problem
\begin{gather*}
u''(t)+\cos(t)=0,\quad t\in [0,1],\\
u(0)=0,\quad u'(1)=-\frac{3}{2} u(\frac{1}{3})
\end{gather*}
has an unique solution
\begin{equation}\label{u2}
u_1(t)= \frac23 t\sin \left( 1 \right) -t\cos \left(\frac13 \right) +t
+\cos \left( t \right) -1,\quad 0\le t\le 1.
\end{equation}
\end{example}

It can be obtained by letting $a=0$, $b=1$, $\eta=\frac{1}{3}$,
$k=-\frac{3}{2}$, $f(t)=\cos(t)$ in Theorem \ref{thm1} that
\begin{equation*}
u_1(t)=\int^1_0B(t,s) \cos(s) ds-t\int^1_0B(\frac{1}{3},s) \cos(s) ds,
\end{equation*}
where $B(t,s)=\min\{t,s\}$, $t,s\in [0,1]$.
Therefore, (\ref{u2}) is obtained by direct computation.
Some properties of $u_1(t)$ are shown in the image Figure 1.

\begin{figure}[ht]
\centering
\includegraphics[width=0.4\textwidth]{fig1}
\caption{Graph of $u_1(t)$}
\end{figure}

\section{The Relate Results for Other Boundary Conditions}

In this section, we give the Green's functions   for some boundary
value problems directly via the following table, omitting the
particular of derivation. The proof is similar to that of Theorem
\ref{thm1}. Similarly, the unique solutions of the linear problems
can be denoted by its Green's functions. Some nonlinear boundary
value problems can be transformed into the nonlinear integral
equations the kernel of which are the Green's functions of the
corresponding linear problems.


\renewcommand\arraystretch{1.6}
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{4}{|l|}{Equation: $u''(t)+f(t)=0$, $t\in [a,b]$,  $a<\eta <b$,
$k$ is a constant.}\\
\hline
no.&assume &boundary & Green's function\\
\hline
1&$k (\eta-a)\neq 1$&\eqref{B11}&
${G_1(t,s)=K(t,s)+\frac{k (t-a)}{1-k (\eta-a)}K(\eta,s)}$\\
\hline
2&$k\neq 1$&(\ref{B12})&${G_2(t,s)=K(t,s)+\frac{k}{1-k}K(\eta,s),}$\\
\hline
3&$k\neq 1$&(\ref{B13})&
${G_3(t,s)=K(t,s)+\frac{k (t-a)}{1-k}K_t(\eta,s)},$ \\
\hline
4&&(\ref{B14})&${G_4(t,s)=K(t,s)+k K_t(\eta,s)},$\\
\hline
5&$k\neq 1$&(\ref{B21})&${G_5(t,s)=H(t,s)+\frac{k}{1-k}H(\eta,s)},$\\
\hline
6&$k (b-\eta)\neq -1$&(\ref{B22})&
${G_6(t,s)=H(t,s)-\frac{k (b-t)}{1+k (b-\eta)}H(\eta,s)},$\\
\hline
7&&(\ref{B23})&${G_7(t,s)=H(t,s)+k H_t(\eta,s)},$\\
\hline
8& $k\neq 1$&(\ref{B24})&${G_8(t,s)=H(t,s)-\frac{k (b-t)}{1-k}H_t(\eta,s)},$\\
\hline
\end{tabular}
\vskip 4mm

\noindent where
\begin{equation*}
K(t,s)=\begin{cases}   s-a, & a\le s\le  t\le b \\
 t-a, & a\le t<s\le b;
\end{cases}
 \qquad
K_t(\eta,s)=\begin{cases}
0, &a\le s<\eta,\\
1, &\eta<s\le b; \end{cases}
\end{equation*}
\begin{equation*}
H(t,s)=\begin{cases}
 b-t, & a\le s\le  t\le b \\
 b-s, & a\le t<s\le b; \end{cases}
\qquad
H_t(\eta,s)=\begin{cases}
 -1, & a\le s<\eta,\\
 0, &\eta<s\le b. \end{cases}
\end{equation*}

\section{Applications in Nonlinear Singular Boundary Value Problems}

In this section,we study the iteration process for the following
nonlinear three-point boundary value problem
\begin{equation}\label{E1}
\begin{gathered}
u''+f(t,u)=0, \quad t\in (0,1),\\
u(0)=0,\quad u'(1)=k u(\eta),
\end{gathered}
\end{equation}
with $\eta\in (0,1)$, $k \eta<1$, $f(t,u)$ may be singular at $t=0$
and/or $t=1$.

Concerning the function $f$ we impose the following hypotheses:
\begin{equation}\label{21}
\begin{aligned}
&\hbox{$f(t,u)$ is nonnegative continuous on $(0,1)\times[0,+\infty)$,}\\
&\hbox{$f(t,u)$ is monotone increasing on $u$, for fixed $t\in (0,1)$,}\\
&\hbox{there exist $q\in (0,1)$ such that }\\
&f(t,ru)\ge r^qf(t,u),\quad\forall\ 0<r<1,\ (t,u)\in (0,1)\times[0,+\infty).
\end{aligned}
\end{equation}
Obviously, from (\ref{21}) we obtain
\begin{equation}\label{23}
f(t,\lambda u)\le \lambda^qf(t,u),\quad\forall\ \lambda>1,\ (t,u)\in
(0,1)\times[0,+\infty).
\end{equation}
It is easy to see that if $0<\alpha_i<1$, $a_i(t)$ are nonnegative
continuous on $(0,1)$, for $i=0,1,2,\dots,m$, then
${f(t,u)=\sum^m_{i=1}a_i(t) u^{\alpha_i}}$ satisfy the
condition (\ref{21}).

Concerning the boundary value problem \eqref{E1}, we have following
conclusions.
\begin{theorem}
Suppose the function $f(t,u)$ satisfy the condition (\ref{21}), and
\begin{equation}\label{25}
0<\int^1_0f(t,t)dt<\infty.
\end{equation}
Then the problem \eqref{E1} has an unique solution $w(t)$ in
$D\bigcap C^2(0,1)$, here
$$
D=\left\{x\in C[0,1]\ |\ \exists M_x\ge m_x>0,\hbox{such that}\;
m_x t\le x(t)\le M_x t,\;t\in I\right\}.
$$
Constructing successively the sequence of functions
\begin{equation}\label{27}
h_n(t)=\int^1_0G(t,s)f(s,h_{n-1}(s))ds,\quad  n=1,2,\dots,
\end{equation}
for any initial function $h_0(t)\in D$, then $\{h_n(t)\}$ must
converge to $w(t)$ uniformly on $[0,1]$ and the
rate of convergence is
\begin{equation}\label{O}
\max_{t\in [0,1]}|h_n(t)-w(t)|=O\big(1-N^{q^n}\big),
 \end{equation}
where $0<N<1$, which  depends on initial function $h_0(t)$,
\begin{equation}\label{G}
G(t,s)=B\left( t,s \right)+{\frac {ktB \left( \eta,s \right)
}{1-k\eta}},\quad
B(t,s)=\min\{t,s\},\quad t,s\in [0,1].
\end{equation}
\end{theorem}


\begin{proof}
Let $J=(0,1)$, $I=[0,1]$, $R^+=[0,+\infty)$,
\begin{gather}
P=\{x(t) \big|\;x(t)\in C(I), x(t)\ge 0\}, \notag \\
\label{F}
Fx(t)=\int^1_0G(t,s)f(s,x(s))ds,\quad\forall x(t)\in D.
\end{gather}
It is easy to see that the operator $F\colon D\to P$ is
increasing. By direct verifications we know that if $u\in D$
satisfies
\begin{equation}\label{F=}
u(t)=Fu(t),\quad t\in I,
\end{equation}
then $u\in C^1(I)\bigcap C^2(J)$ is a solution of \eqref{E1}.

For any $x\in D$, there exist positive numbers $0<m_x<1<M_x$ such
that
\begin{gather}
m_x\;s\le x(s)\le M_x\;s,\quad s\in I, \notag\\
\label{fs}
(m_x)^qf(s,s)\le f(s,x(s))\le (M_x)^q f(s,s),\quad s\in J.
\end{gather}
By (\ref{G}) we have
\begin{gather}\label{G>}
G(t,s)=B(t,s)+\frac{k t}{1-k \eta}B(\eta,s)
\ge t \frac{k}{1-k \eta}B(\eta,s), \\
\label{G<}
G(t,s)\le t+\frac{k t}{1-k \eta}B(\eta,s) \le
t\left(1+\frac{k}{1-k \eta}B(\eta,s)\right).
\end{gather}
Using  (\ref{F}), (\ref{23}), (\ref{fs}), (\ref{G>}), (\ref{G<})
 and the conditions (\ref{21}), we obtain
\begin{equation}\label{29}
 Fx(t)\ge t (m_x)^q\frac{k}{1-k \eta}\int^1_0B(\eta,s)f(s,s)ds, \quad
t\in I,
\end{equation}
\begin{equation}\label{31}
\begin{aligned}
Fx(t)&=\int^1_0G(t,s)f(s,x(s))ds\\
&\le  t(M_x)^q \int^1_0\left(1+\frac{k}{1-k \eta}B(\eta,s)\right)f(s,s)ds
, \quad t\in I.
\end{aligned}
\end{equation}
Equations (\ref{25}), (\ref{29}) and (\ref{31}) imply that
$F\colon D\to D$.

For any $h_0\in D$, we let
\begin{equation}\label{32}
\begin{gathered}
l_{h_0}=\sup\left\{l>0 : l h_0(t)\le (Fh_0)(t),\;t\in I\right\},\\
L_{h_0}=\inf\left\{L>0 : (Fh_0)(t))\le L h_0(t),\;t\in I\right\}, \\
m=\min\{1, (l_{h_0})^{\frac{1}{1-q}}\},\quad
M=\max\{1, (L_{h_0})^{\frac{1}{1-q}}\},
\end{gathered}
\end{equation}
\begin{equation}\label{34}
\begin{gathered}
u_0(t)=m h_0(t),\quad  v_0(t)=M h_0(t),\\
u_n(t)=Fu_{n-1}(t),\quad  v_n(t)=Fv_{n-1}(t),\quad n=0,1,2,\dots.
\end{gathered}
\end{equation}
Since the operator $F$ is increasing, (\ref{21}), (\ref{32}) and
(\ref{34}) imply
\begin{equation}\label{35}
u_0(t)\le u_1(t)\le\dots\le u_n(t)\dots\le v_n(t)\le\dots\le
v_1(t)\le v_0(t),\quad t\in I.
\end{equation}
For $t_0=m/M$, from (\ref{F}), (\ref{21}) and (\ref{34}), it can
obtained by induction that
\begin{equation}\label{45}
u_n(t)\ge (t_0)^{q^n}v_n(t),\quad t\in I,\; n=0,1,2,\dots.
\end{equation}
 From (\ref{35}) and (\ref{45}) we know that
\begin{equation}\label{46}
0\le u_{n+p}(t)-u_n(t)\le
v_n(t)-u_n(t)\le\big(1-(t_0)^{q^n}\big)M h_0(t), \forall n,p ,
\end{equation}
so that there exists  a function $w(t)\in D$ such that
\begin{gather}\label{47}
u_n(t)\to w(t),\quad v_n(t)\to w(t),\quad(\hbox{uniformly on $I$}), \\
\label{48}
u_n(t)\le w(t)\le v_n(t),\quad t\in I,\; n=0,1,2,\dots.
\end{gather}
 From the operator $F$ being increasing and (\ref{34}) we have
$$
u_{n+1}(t)=Fu_n(t)\le Fw(t)\le Fv_n(t)=v_{n+1}(t),\quad
n=0,1,2,\dots.
$$
This together with (\ref{47}) and uniqueness of the limit imply that
$w(t)$ satisfy (\ref{F=}), hence $w(t)\in C^1(I)\bigcap C^2(J)$ is a
solution of \eqref{E1}.

 Form (\ref{27}) (\ref{34}) and the operator $F$ being increasing, we obtain
\begin{equation}\label{49}
u_n(t)\le h_n(t)\le v_n(t),\quad t\in I,\; n=0,1,2,\dots,
\end{equation}
thus, it follows from (\ref{46}), (\ref{48}) and (\ref{49}) that
\begin{align*}
|h_n(t)-w(t)|&\le|h_n(t)-u_n(t)|+|u_n(t)-w(t)|\\
&\le 2|v_n(t)-u_n(t)|\\
&\le \big(1-(t_0)^{q^n}\big)M |h_0(t)|.
\end{align*}
Therefore,
\begin{equation*}
\max_{t\in I}|h_n(t)-w(t)|\le
\big(1-(t_0)^{q^n}\big)M \max_{t\in I}|h_0(t)|.
\end{equation*}
So that (\ref{O}) holds. From $h_0(t)$ is arbitrary in $D$
we know that $w(t)$ is the unique solution of the equation
(\ref{F=}) in $D$.
\end{proof}

\begin{remark} \label{rmk2} \rm
If $f(t,u)$ is continuous on $I\times R^+$, then it is quite evident
that the condition (\ref{25}) holds. Hence the unique solution
$w(t)$ is in $C^2(I)$.
\end{remark}

\begin{thebibliography}{00}

\bibitem{bai1}
C. Bai, J. Fang; \emph{Existence of positive solutions for
three-point boundary value problems at resonance},
J. Math. Anal. Appl., 291(2004),538--549.

\bibitem{bai}
Z.Bai, W.Ge, Existence of three positive solutions for some second-order boundary
 value problems, Computers and Mathematics with Applications, 48 (2004), 699-707.

\bibitem{wingsum}
W. Cheung, J. Ren; \emph{Positive solution for m-point boundary value problems},
J. Math. Anal. Appl., 303 (2005), 565-575.

\bibitem{guo88}
D. Guo, V. Lakshmikantham;
\emph{Nonlinear Problems  in Abstract Cones},  Academic Press, Boston, 1988.

\bibitem{Yanping04}
Y. Guo, W. Ge; \emph{Positive solutions for three-point boundary value problems with
dependence on the first order derivative},
J. Math. Anal. Appl. 290(2004), 291--301.

\bibitem{Lee}
S. N. Ha, C. R. Lee; \emph{Numerical study for two-point boundary value
problems using Green's functions}.
Computers and Mathematics with Applications 44(2002),1599-1608.

\bibitem{bong}
B. H. Im, E. K. Lee, Yong-Hoon Lee;
\emph{A global bifurcation phenomenon for second order
singular boundary value problems}, J. Math. Anal. Appl., 308 (2005), 61-78.

\bibitem{kaufmann}
E. R. Kaufmann, N. Kosmatov;
\emph{A Second-Order Singular Boundary Value Problem}, Computers and
Mathematics with Applications 47 (2004), 1317-1326.

\bibitem{Webb}
R. A. Khan, J. R. L. Webb;
\emph{Existence of at least three solutions of a second-order
three-point boundary value problem}, Nonlinear Analysis 64(2006), 1356--1366.

\bibitem{Bing06}
B. Liu; \emph{Positive solutions of second-order three-point boundary value problems
with change of sign in Banach spaces}, Nonlinear Analysis, 64(2006),1336--1355.

\bibitem{bing1}
B. Liu; \emph{Positive solutions of singular three-point boundary value problems for the
 one-dimensional p-Laplacian}, Computers and Mathematics with Applications, 48(2004),913--925.

\bibitem{BingCMA}
B. Liu; \emph{Positive solutions of second-order three-point boundary
value problems with change of sign},
Computers and Mathematics with Applications, 47(2004), 1351--1361.

\bibitem{bing2}
B. Liu; \emph{Positive solutions of three-point boundary value problems for
the one-dimensional p-Laplacian with infinitely many
singularities}, Applied Mathematics Letters, 17(2004),655--661.

\bibitem{ruyun}
R. Ma, B. Thompson;
\emph{Global behavior of positive solutions of nonlinear three-point boundary
 value problems}, Nonlinear Analysis 60 (2005), 685- 701.

\bibitem{Pokornyi}
Yu. V. Pokornyi, A. V.Borovskikh;
\emph{The connection of the Green's function and the influence
function for nonclassical problems}, Journal of Mathematical Sciences, 119(6),(2004) 739-768.

\bibitem{singh}
P. Singh; \emph{A second-order singular three-point boundary value problem},
Applied Mathematics Letters, 17(2004), 969-976.

\bibitem{yps}
Y. Sun, L. Liu; \emph{Solvability for a nonlinear second-order
three-point boundary value problem}, J. Math. Anal. Appl., 296(2004), 265--275.

\bibitem{xan05}
X. Xu; \emph{Three solutions for three-point boundary value problems},
Nonlinear Analysis 62(2005), 1053--1066.

\bibitem{XianSun04}
X. Xu, J. Sun; \emph{On sign-changing solution for some three-point boundary
 value problems}, Nonlinear Analysis, 59(2004), 491--505.

\bibitem{Yang}
C. Yang, C. Zhai, Jurang Yan;
\emph{Positive solutions of the three-point boundary
value problem for second order differential equations with
an advanced argument}, Nonlinear Analysis, In Press.

\end{thebibliography}

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