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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**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}
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}
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\end{document}
**