\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 41, pp. 1--7.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/41\hfil Existence of positive solutions]
{Existence of positive solution for semipositone second-order
 three-point boundary-value problem}

\author[J.-P. Sun, J. Wei\hfil EJDE-2008/41\hfilneg]
{Jian-Ping Sun, Jia Wei}  % in alphabetical order

\address{Jian-Ping Sun \newline
Department of Applied Mathematics, Lanzhou University of Technology\\
Lanzhou, Gansu, 730050,  China}
\email{jpsun@lut.cn}

\address{Jia Wei \newline
Department of Applied Mathematics, Lanzhou University of Technology\\
Lanzhou, Gansu, 730050,  China}
\email{weijia\_vick@163.com}

\thanks{Submitted August 28, 2007. Published March 20, 2008.}
\subjclass[2000]{34B15} 
\keywords{Semipositone; boundary value problem; positive solution; 
\hfill\break\indent  existence; fixed-point}

\begin{abstract}
 In this paper, we establish the existence of positive solution for
 the semipositone second-order three-point boundary value problem
 $u''(t)+\lambda f(t,u(t))=0$, $0<t<1$,
 $u(0)=\alpha u(\eta)$, $ u(1)=\beta u(\eta)$.
 Our arguments are based on the well-known Guo-Krasnosel'skii
 fixed-point theorem in cones.
\end{abstract}

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

\section{Introduction}

Multi-point boundary value problems (BVPs for short), due to their
 applications to almost all areas of science, engineering
and technology, have attracted considerable attention. For example,
in 1987, Il'in and Moiseev \cite{11} studied some multi-point BVPs
first for linear second-order ordinary differential equations, and
then, many authors discussed nonlinear multi-point BVPs, see
\cite{g2,g3,m1,m2,s1,y1,y2} and the references therein. In particular, Ma
\cite{m1} showed the existence of at least one positive solution for
the  three-point BVP
\begin{gather*}
u''(t)+h(t)f(u(t))=0,\quad 0<t<1, \\
u(0)=0,\quad u(1)=\alpha u(\eta),
\end{gather*}
under the condition that $0<\alpha\eta<1$ and $f$ was nonnegative.

Recently, when the nonlinear term is not necessarily  nonnegative, Yao
\cite{y1} proved the existence of at least one positive solution for
the three-point BVP
\begin{gather*}
u''(t)+\lambda f(t,u(t))=0,\quad 0<t<1, \\
u(0)=0,\quad u(1)=\alpha u(\eta),
\end{gather*}
where $0<\alpha\eta<1$ and $\lambda>0$ was a parameter.

Motivated by the excellent results in \cite{m1,y1},  we
are concerned with the existence of positive solution for the
 second-order three-point BVP
\begin{gather}
u''(t)+\lambda f(t,u(t))=0,\quad 0<t<1,  \label{e1.1} \\
u(0)=\alpha u(\eta),\quad u(1)=\beta u(\eta),  \label{e1.2}
\end{gather}
where $0<\eta<1$, $0<\beta\leq\alpha<1$, $\lambda>0$ is a parameter.
Throughout, we assume that there exists a constant $M>0$ such that
$f:[0,1]\times [0,+\infty )\to (-M,+\infty )$ is continuous.
This implies that the BVP \eqref{e1.1} and \eqref{e1.2} is semipositone.
For convenience, we denote
\begin{gather*}
\xi=1-\alpha+(\alpha-\beta) \eta,\\
\gamma=\min\big\{\frac{\alpha\eta}{1-\alpha+\alpha\eta},\frac{(1-\eta)
\alpha}{1-\beta \eta}\big\},
\\
B=\max\{f(t,u)+M : (t,u)\in [0,1]\times [0,1]\}.
\end{gather*}

The main result of this paper is the following theorem.

\begin{theorem}\label{t1}
Suppose that
$\lim_{u\to +\infty}\min_{0\leq t \leq\eta}\frac{f(t,u)}{u}=+\infty$.
Then the BVP \eqref{e1.1} and \eqref{e1.2} has at least one positive
solution for
\[
0<\lambda <\min \big\{ \frac{2\xi }{B(1-\alpha +\alpha \eta )},\ \frac{%
2\gamma \beta \xi }{\alpha M(1-\alpha +\alpha \eta -\beta \eta
^2)}\big\}.
\]
\end{theorem}

Our main tool is the well-known Guo-Krasnosel'skii fixed-point
theorem, which we state here for convenience of the reader.

\begin{theorem}[\cite{g1,k1}]\label{t2}
Let $E$ be a Banach space, $K$ a cone in $E$ and
$\Omega _c=\{u\in K:\ \parallel u\parallel <c\}$.
Suppose that $T:K\to K$ is a completely continuous operator and
$0<a<b<+\infty $ such that either
\begin{itemize}
\item[(1)] $Tu\nleqslant u$ for $u\in \partial \Omega _a$ and
$u\nleqslant Tu$ for $u\in \partial \Omega _b$, or

\item[(2)] $u\nleqslant Tu$ for $u\in \partial \Omega _a$ and
$Tu\nleqslant u$ for $u\in \partial \Omega _b$.
\end{itemize}
Then $T$ has a fixed point in $\overline{\Omega}_{b}\setminus\Omega_{a}$.
\end{theorem}

\section{Preliminaries}

In the remainder of this paper, we  assume that
$0<\beta\leq\alpha<1$. Also let the Banach space $E=C[0,1]$ be
equipped with the usual norm $\|u\|=\max_{t\in [0,1]}|u(t)|$.

\begin{lemma} \label{lem2.1}
For any fixed $y\in E$, the BVP
\begin{gather}
u''(t)+y(t)=0,\quad 0<t<1, \label{e2.1}\\
u(0)=\alpha u(\eta ),\quad u(1)=\beta u(\eta ) \label{e2.2}
\end{gather}
has a unique solution
\begin{align*}
u(t)&=-\int_0^t(t-s)y(s)ds+\frac 1\xi [(1-\alpha )t+\alpha \eta
]\int_0^1(1-s)y(s)ds \\
&\quad +\frac 1\xi [(\alpha -\beta )t-\alpha]\int_0^\eta (\eta -s)y(s)ds.
\end{align*}
\end{lemma}

Since the proof of the above lemma is easy, we omit it.

\begin{lemma}\label{l2}
If $y\in E$ and $y\geq 0$, then the unique solution $u$ of the BVP
\eqref{e2.1}--\eqref{e2.2} satisfies $ u(t)\geq 0$ for $t\in [0,1]$.
\end{lemma}

\begin{proof} Since
$u''(t)=-y(t)\leq0$, $0<t<1$ it follows that the graph of $u(t)$ is
concave dawn, we only need to prove $u(0)\geq0$ and $u(1)\geq0$. In
view of $0<\beta \leq \alpha <1$ and (2.2), we know that $u(0)$,
$u(\eta)$ and $u(1)$ have same signs. Suppose on the contrary that
$u(0)<0$, $u(\eta)<0$ and $u(1)<0$. Then we have
\[
u(\eta)=\frac{u(0)}{\alpha}<u(0), \quad
u(\eta)=\frac{u(1)}{\beta}<u(1)\,.
\]
Then
\[
u(\eta)<\min\{u(0),\ u(1)\},
\]
which contradicts the concavity of $u$. Thus, we get that
\[
u(0)\geq 0\quad \text{and}\quad u(1)\geq 0
\]
as required.
\end{proof}

\begin{lemma}\label{l3}
If $y\in E$ and $y\geq 0$, then the unique solution $u$ of the BVP
\eqref{e2.1}--\eqref{e2.2} satisfies
\begin{equation} \label{e2.3}
\min_{0\leq t\leq\eta}u(t)\geq\gamma\| u\|.
\end{equation}
\end{lemma}

\begin{proof}
Since $u(0)=\alpha u(\eta )$, $0< \alpha <1$ and
Lemma \ref{l2} imply that $u(0)\leq u(\eta )$, we know that
\begin{equation}
\min_{0\leq t\leq \eta }u(t)=u(0).  \label{e2.4}
\end{equation}
Set $u(\overline{t})=\| u\|$. We consider the following two cases:

Case 1. $\eta \leq \overline{t}$. It follows from the concavity of
$u$ that
\[
\frac{u(\eta)-u(0)}{\eta-0}\geq \frac{u(\overline{t})-u(0)}{\overline{t}-0}.
\]
Combining the boundary condition $u(0)=\alpha u(\eta)$, we conclude
that
\[
u(0)\geq
\frac{\alpha\eta}{1-\alpha+\alpha\eta}u(\overline{t})
=\frac{\alpha\eta}{1-\alpha+\alpha\eta}\| u\|,
\]
which together with \eqref{e2.4} implies
\begin{equation} \label{e2.5}
\min_{0\leq t\leq\eta}u(t)\geq
\frac{\alpha\eta}{1-\alpha+\alpha\eta} \| u\|.
\end{equation}

Case 2. $\overline{t}<\eta $. It follows from the concavity of $u$
that
\[
u(\overline{t})\leq \frac{u(1)-u(\eta)}{1-\eta}(0-\eta) +u(\eta),
\]
which together with \eqref{e2.4} and the boundary conditions
$u(0)=\alpha u(\eta)$ and $u(1)=\beta u(\eta)$ implies
\begin{equation}
\min_{0\leq t\leq\eta}u(t)\geq \frac{(1-\eta)\alpha}{1-\beta\eta}\|
u\|. \label{e2.6}
\end{equation}
By (\ref{e2.5}) and (\ref{e2.6}), we know that (\ref{e2.3}) is fulfilled.
\end{proof}

\begin{lemma}
The BVP
\begin{gather}
\widetilde{u}''(t)+1=0,\quad 0< t< 1, \label{e2.7}\\
\widetilde{u}(0)=\alpha \widetilde{u}(\eta),\quad
 \widetilde{u}(1)=\beta
\widetilde{u}(\eta) \label{e2.8}
\end{gather}
has a unique solution
\[
\widetilde{u}(t)=-\frac{t^2}2+\frac{(1-\alpha )t+\alpha \eta
+[(\alpha -\beta )t-\alpha ]\eta ^2}{2\xi },\quad t\in [0,1].
\]
\end{lemma}

\begin{remark} \label{r1} \rm
The unique solution $ \widetilde{u}$ of the BVP
\eqref{e2.7}--\eqref{e2.8} satisfies
\[
\widetilde{u}(t)\leq \frac{1-\alpha +\alpha \eta -\beta \eta
^2}{2\xi },\quad t\in [0,1].
\]
\end{remark}

\section{Proof of Theorem \ref{t1}}

Let
\begin{gather*}
g(t,u)=f(t,u)+M,\quad (t,u)\in [0,1]\times [0,+\infty),\\
\overline{g}(t,u)=g(t,\max\{u,0\}),\quad (t,u)\in [0,1]\times
(-\infty,+\infty).
\end{gather*}
Obviously, $\overline{g}:[0,1]\times (-\infty,+\infty)\to
(0,+\infty) $ is continuous.
We consider the  BVP
\begin{gather} \label{e3.1}
u''(t)+\lambda \overline{g}(t,u(t)-w(t))=0,\quad 0<t<1,\\
u(0)=\alpha u(\eta),\quad u(1)=\beta u(\eta), \label{e3.2}
\end{gather}
where $w(t)=\lambda M\widetilde{u}(t)$ and $\widetilde{u}(t)$ is the
solution of the BVP \eqref{e2.7}--\eqref{e2.8}. It is not difficult
to prove that $u^{\ast}$ is a positive solution of the BVP
\eqref{e1.1}--\eqref{e1.2} if and only if $\overline{u}=u^{\ast}+w$
is a solution of the BVP \eqref{e3.1}--\eqref{e3.2} and
$\overline{u}(t)>w(t)$, $0<t<1$.

We define an operator $T_\lambda :E\to E$:
\begin{align*}
(T_\lambda u)(t)
&=-\lambda \int_0^t(t-s)\overline{g}(s,u(s)-w(s))ds\\
&\quad +\frac \lambda \xi [(1-\alpha )t+\alpha \eta ]\int_0^1(1-s)
\overline{g} (s,u(s)-w(s))ds \\
&\quad +\frac \lambda \xi [(\alpha -\beta )t-\alpha ]
\int_0^\eta (\eta -s) \overline{g}(s,u(s)-w(s))ds,\quad t\in [0,1].
\end{align*}
It is easy to check that $\overline{u}\in E$ is a solution of the
BVP \eqref{e3.1}--\eqref{e3.2} if and only if $\overline{u}$ is a
fixed point of the operator $ T_{\lambda}$ in $E$. Therefore, we
only need to prove that the operator $T_{\lambda}$ has a fixed point
$\overline{u}\in E$ and $ \overline{u}(t)>w(t)$, $0<t<1$. Denote
\[
K=\{ u\in E:\min_{0\leq t\leq 1}u(t)\geq 0,\;
\min_{0\leq t\leq \eta }u(t)\geq \gamma \Vert u\Vert \}.
\]
Obviously, $K$ is a cone in $E$. It follows from Lemma \ref{l3} that
$T_{\lambda}K\subset K$. Furthermore, we can prove that
$T_{\lambda}:K\to K$ is completely continuous.
Now, we introduce a partial order in $E$. Let $x_{1}$, $x_{2}\in E$.
We say $x_{1}\leq x_{2}$ if and only if $x_{2}-x_{1}\in K$.

If we let $\Omega_{1}=\{u\in K: \|u\|<1\}$, then we may assert that
\begin{equation}
u\nleqslant T_{\lambda}u \quad \text{for any }
u\in\partial\Omega_{1}.\label{e9}
\end{equation}
Suppose on the contrary that there exists a
$u_{0}\in\partial\Omega_{1}$ such that $u_{0}\leq T_{\lambda}u_{0}$.
Since $u_{0}(t)-w(t)\leq 1$ and $(\alpha-\beta)t-\alpha<0$,
 $0\leq t\leq1$, we have
\begin{align*}
u_0(t)&\leq (T_\lambda u_0)(t) \\
&=-\lambda \int_0^t(t-s)\overline{g} (s,u_0(s)-w(s))ds \\
&\quad +\frac\lambda \xi [(1-\alpha )t+\alpha \eta ]\int_0^1(1-s)\overline{g}
(s,u_0(s)-w(s))ds \\
&\quad+\frac \lambda \xi [(\alpha -\beta )t-\alpha ]\int_0^\eta (\eta
-s)\overline{g}(s,u_0(s)-w(s))ds \\
&\leq \frac \lambda \xi [(1-\alpha )t+\alpha \eta
]\int_0^1(1-s)\overline{g}
(s,u_0(s)-w(s))ds \\
&\leq \frac \lambda \xi [(1-\alpha )t+\alpha \eta ]\int_0^1(1-s)\max_{0\leq
s\leq 1}\overline{g}(s,u_0(s)-w(s))ds \\
&=\frac \lambda \xi [(1-\alpha )t+\alpha \eta ]\int_0^1(1-s)\max_{0\leq
s\leq 1}\left[ f(s,\max\{u_0(s)-w(s),0\})+M\right] ds \\
&\leq \frac{B\lambda }{2\xi }[(1-\alpha )t+\alpha \eta ],\ t\in
[0,1],
\end{align*}
which leads to a contradiction:
\[
1=\|u_{0}\|\leq \frac{B\lambda}{2\xi}(1-\alpha+\alpha\eta) <1.
\]
So, \eqref{e9} is satisfied.

On the other hand, we  claim that there exists a constant
$\sigma>1$ such that
\begin{equation}
T_{\lambda}u\nleqslant u\quad  \text{for any }
u\in\partial\Omega_{\sigma}.\label{e10}
\end{equation}

In fact, if we let $V_{\lambda}=\{u\in K:T_{\lambda}u\leq u\}$ and
$m_{\lambda}=\sup\{\|u\|:u \in V_{\lambda}\}$, then we only need to
prove $m_{\lambda}<+\infty$. Suppose on the contrary that there
exists a sequence $\{u_{n}\}_{n=1}^{\infty}\subset K$ such that
$T_{\lambda}u_{n}\leq u_{n}$ and $\|u_{n}\|\to +\infty$
($n\to +\infty$). Then for any $t\in [0,\eta]$, we have
\begin{equation}
u_{n}(t)-w(t)\geq\gamma\| u_{n}\|-\| w\|\to+\infty\
\quad (n\to +\infty).\label{e11}
\end{equation}
In view of (\ref{e11}) and $\lim_{u\to +\infty }\min_{0\leq
t\leq \eta }\frac{f(t,u)} u=+\infty $, we know that
\begin{equation}
\lim_{n\to +\infty }\min_{0\leq t\leq \eta
}\frac{\overline{g} (t,u_n(t)-w(t))}{u_n(t)-w(t)}=+\infty
.\label{e12}
\end{equation}
So, there exists a positive integer $N$ such that for any $n\geq N$,
\begin{equation}
\min_{0\leq t\leq \eta }[u_n(t)-w(t)]\geq \frac \gamma 2\| u_n\|
\label{e13}
\end{equation}
and
\begin{equation}
\min_{0\leq t\leq \eta
}\frac{\overline{g}(t,u_n(t)-w(t))}{u_n(t)-w(t)}\geq \frac{4\xi
}{\lambda \gamma }\Big[ (1-\eta )\int_0^\eta tdt\Big]
^{-1}.\label{e14}
\end{equation}
For the rest of this article, we let $n\geq N$.
Noticing $T_\lambda u_n\in K$, we
have $0\leq (T_\lambda u_n)(t)\leq u_n(t)$, $t\in [0,1]$. And so,
\begin{equation}
\| u_{n}\| =\max_{0\leq t\leq 1}u_n(t)\geq \max_{0\leq t\leq
1}(T_\lambda u_n)(t)\geq (T_\lambda u_n)(\eta ).\label{e15}
\end{equation}
At the same time, by (\ref{e13}) and (\ref{e14}), we also obtain
\begin{align*}
&(T_\lambda u_n)(\eta )\\
&=-\lambda \int_0^\eta (\eta
-s)\overline{g}(s,u_n(s)-w(s))ds+\frac \lambda
\xi \eta \int_0^1(1-s)\overline{g}(s,u_n(s)-w(s))ds \\
&\quad +\frac \lambda \xi [(\alpha -\beta )\eta -\alpha ]\int_0^\eta (\eta -s)%
\overline{g}(s,u_n(s)-w(s))ds \\
&=\frac \lambda \xi (1-\eta )\int_0^\eta s\overline{g}(s,u_n(s)-w(s))ds+%
\frac \lambda \xi \eta \int_\eta ^1(1-s)\overline{g}(s,u_n(s)-w(s))ds \\
&\geq \frac \lambda \xi (1-\eta )\int_0^\eta s\overline{g}(s,u_n(s)-w(s))ds
\\
&\geq \frac \lambda \xi (1-\eta )\int_0^\eta s\min_{0\leq s\leq
\eta }\big[\frac{\overline{g}(s,u_n(s)-w(s))}{u_n(s)-w(s)}\big]
\min_{0\leq s\leq \eta }[u_n(s)-w(s)]ds \\
&\geq \frac \lambda \xi (1-\eta ) \frac{4\xi }{\lambda \gamma
}\Big[ (1-\eta )\int_0^\eta tdt\Big] ^{-1}  \frac \gamma 2\|
u_{n}\|\cdot  \int_0^\eta sds  \\
&=2\| u_{n}\|,
\end{align*}
which together with (\ref{e15}) implies
\[
\|u_{n}\|\geq(T_{\lambda}u_{n})(\eta)\geq2\| u_{n}\|.
\]
This is impossible. So, $m_{\lambda}<+\infty$. And so, \eqref{e10}
is fulfilled.

It follows from \eqref{e9}, \eqref{e10} and Theorem \ref{t2} that
$T_\lambda $ has a fixed point $\overline{u}\in \overline{\Omega
}_\sigma \setminus \Omega _1$. With the similar arguments as in
Lemma \ref{l3}, we know that
\[
\min_{0\leq t\leq 1}\overline{u}(t)=\overline{u}(1)=\frac \beta
\alpha \overline{u}(0)\geq \frac{\beta \gamma}{\alpha} \|
\overline{u}\|,
\]
which together with Remark \ref{r1} implies
\[
\overline{u}(t) \geq \frac{\beta \gamma}{\alpha} \| \overline{u}
\| \geq\frac{\beta \gamma}{\alpha}
>\lambda M\cdot \frac{1-\alpha +\alpha \eta -\beta \eta ^2}{2\xi }
\geq \lambda M\cdot \widetilde{u}(t) =w(t),
\]
for $t\in (0,1)$. Therefore, $u^{\ast}= \overline{u}-w$ is a
positive solution of the BVP \eqref{e1.1}--\eqref{e1.2}.

\begin{thebibliography}{00}


\bibitem{g1}  D. Guo, V. Lakshikantham, Nonlinear Problems in
Abstract Cones, Academic Press, San Diego, 1988.

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

\bibitem{g3}  C. P. Gupta, S. Trofimchuk, Existence of a solution
of a three-point boundary value problem and the spectral radius of a
related linear operator, Nonlinear Anal. 34 (1998) 489-507.

\bibitem{11}  V. A. Il'in, E. I. Moiseev, Nonlocal boundary value
problem of the second kind for a Sturm-Liouville operator,
Differential Equations 23 (1987) 979-987.

\bibitem{k1}  M. A. Krasnosel'skii, Positive Solutins of Operator
Equations, Noordhoff, Groningen, The Netherlands, 1964.

\bibitem{m1}  R. Ma, Positive solutions of a nonlinear three-point
boundary value problems, Electronic J. Differential Equations 34
(1999) 1-8.

\bibitem{m2}  R. Ma, Multiplicity of positive solutions for
second-order three-point boundary value problems, Comput. Math.
Appl. 40 (2000) 193-204.

\bibitem{s1}  J. P. Sun, W. T. Li, Y. H. Zhao, Three positive
solutions of a nonlinear three-point boundary value problem, J. Math. Anal.
Appl. 288 (2003) 708-716.

\bibitem{y1}  Q. Yao, An existence theorem of positive solution for
a superlinear semipositone second-order three-point BVP, J.
Mathematical Study 35 (2002) 32-35.

\bibitem{y2}  Q. Yao, R. Ma, Multiplicity of positive solutions
for second-order three-point boundary value problems, Comput. Math. Appl.
l40 (2000) 193-204.



\end{thebibliography}

\end{document}