\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 152, 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}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2012/152\hfil Multiple positive solutions]
{Multiple positive solutions for second-order three-point
boundary-value problems with sign changing nonlinearities}
\author[J. Liu, Z. Zhao \hfil EJDE-2012/152\hfilneg]
{Jian Liu, Zengqin Zhao} % in alphabetical order
\address{Jian Liu \newline
School of Mathematics and Quantitative Economics,
Shandong University of Finance and Economics,
Jinan, Shandong, 250014, China}
\email{liujianmath@163.com}
\address{Zengqin Zhao \newline
School of Mathematical Sciences, Qufu Normal
University, Qufu, Shandong, 273165, China}
\email{zqzhao@mail.qfnu.edu.cn}
\thanks{Submitted March 14, 2012. Published September 7, 2012.}
\subjclass[2000]{34B15, 34B25}
\keywords{Multiple positive solutions; sign changing; fixed-point theorem}
\begin{abstract}
In this article, we study the second-order three-point boundary-value
problem
\begin{gather*}
u''(t)+a(t)u'(t)+f(t,u)=0,\quad 0 \leq t \leq 1, \\
u'(0)=0,\quad u(1)=\alpha u(\eta),
\end{gather*}
where $0<\alpha$, $\eta<1$, $a\in C([0,1],(-\infty, 0))$ and $f$ is
allowed to change sign. We show that there exist two positive
solutions by using Leggett-Williams fixed-point theorem.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks
\section{Introduction}
The study of multi-point boundary-value problems for
linear second-order ordinary differential equations was initiated
by Kiguradze and Lomtatidze \cite{Kiguradze}, Lomtatidze \cite{Lomtatidze},
Il'in and Moviseev \cite{Il'in1,Il'in2}, Agarwal and Kiguradze \cite{AK},
Lomtatidze and Malaguti \cite{LM}. Motivated by the study of \cite{Il'in1,Il'in2},
Gupta \cite{Gupta1} 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. We refer the reader to Gupta \cite{Gupta2},
Li, Liu and Jia \cite{LiLiuJia}, Liu \cite{Liu}, Ma\cite{Ma1,Ma2} for some
references along this line. Using results from fixed point theory,
such as the fixed-point theorems by Banach, Krasnosel'skii,
Leggett-Williams etc., in studying second-order dynamic systems is a standard
and useful tool (see, e.g. \cite{Anderson2,Anderson1,BS}).
Recently, Ma \cite{Ma1} studied the three-point boundary-value problem
(BVP)
\begin{gather*}
u''(t)+a(t)f(u)=0,\quad 0 \leq t \leq 1, \\
u(0)=0,\quad u(1)=\alpha u(\eta),
\end{gather*}
where $0<\eta<1$, $\alpha$ is a positive constant, $a\in C[0,1]$,
$f\in C([0,+\infty),[0,+\infty))$ and there exists $x_0\in(0,1)$
such that $a(x_0)>0$. Author got the existence and multiplicity
of positive solutions theorems under the condition that $f$ is
either superlinear or sublinear by using Krasnoselskii's fixed
point theorem.
In 2001, Ma\cite{Ma2} considered $m$-point boundary-value problem
\begin{gather*}
u''(t)+h(t)f(u)=0,\quad 0 \leq t \leq 1, \\
u(0)=0,\quad u(1)=\sum _{i=1}^{m-2}\beta_{i} u(\xi_{i}),
\end{gather*}
where
$\beta_{i}>0$ ($i=1,2,\dots,m-2$), $0<\xi_{1}<\xi_{2}<\dots<\xi_{m-2}<1$,
$h\in C([0,1],[0,+\infty))$ and $f\in C([0,+\infty),[0,+\infty))$.
Author established the existence of positive solutions
under the condition that $f$ is either superlinear or sublinear.
In \cite{LiLiuJia}, the authors studied the three-point boundary-value problem
\begin{gather*}
u''(t)+a(t)u'(t) +\lambda f(t, u)=0,\quad 0 \leq t \leq 1, \\
u'(0)=0,\quad u(1)=\alpha u(\eta),
\end{gather*}
where $0<\eta<1$, $\alpha$ is a positive constant,
$a\in C([0,1],(-\infty, 0))$, $ f\in C([0,1]\times \mathbb{R^+},\mathbb{R})$
and there exists $M>0$ such that $f(t,u)\geq -M$ for
$(t,u)\in [0,1]\times \mathbb{R^+} $.
They obtained the existence of one positive solution by using
Krasnoselskii's fixed point theorem.
Motivated by the results mentioned above, in this paper, we study
the existence of positive solutions of three-point boundary-value problem
\begin{equation}
\begin{gathered}
u''(t)+a(t)u'(t)+f(t,u)=0,\quad 0 \leq t \leq 1, \\
u'(0)=0,\quad u(1)=\alpha u(\eta),
\end{gathered} \label{e1.1}
\end{equation}
where $0<\alpha$, $\eta<1$, $a\in C([0,1],(-\infty, 0))$ and $f$ is
allowed to change sign. We show that there exist two positive
solutions by using Leggett-Williams fixed-point theorem. Our ideas
are similar those used in \cite{LiLiuJia}, but a little different.
By applying Leggett-Williams fixed-point theorem, we get the new
results, which are different from the previous results and the
conditions are easy to be checked. In particular, we do not need
that $f$ be either superlinear or sublinear which was required in
\cite{Gupta2,LiLiuJia,Ma1,Ma2}.
In the rest of the paper, we make the following assumptions
\begin{itemize}
\item[(H1)] $0<\alpha$, $\eta<1$;
\item[(H2)] $a\in C([0,1],(-\infty, 0))$;
\item[(H3)] $f: [0,1]\times \mathbb{R^+}\to \mathbb{R}$ is continuous and
there exists $M>0$ such that $f(t,u)\geq -M$ for
$(t,u)\in [0,1]\times \mathbb{R^+}$.
\end{itemize}
By a positive solution of \eqref{e1.1}, we understand a function $u$
which is positive on $(0,1)$ and satisfies the differential
equations as well as the boundary conditions in \eqref{e1.1}.
\section{Preliminaries}
In this section, we give some definitions and lemmas.
\begin{definition} \label{def2.1} \rm
Let $E$ be a real Banach space. A nonempty closed set $P\subset E$
is said to be a cone provided that
\begin{itemize}
\item[(i)] $u\in P$ and $a\geq0$ imply $a u\in P$;
\item[(ii)] $u,\ -u \in P$ implies $u=0$;
\item[(ii)] $u,v\in P$ implies $u+v\in P$.
\end{itemize}
\end{definition}
\begin{definition} \label{def2.2}\rm
Given a cone $P$ in a real Banach space $E$, an operator $\psi: P\to P$ is said to be
increasing on $P$, provided $\psi(x)\leq\psi(y)$, for all
$x,y\in P$ with $x\leq y$.
A functional $\alpha: P\to [0,\infty)$ is said
to be nonnegative continuous concave on $P$, provided
$\alpha(tx+(1-t)y)\geq t\alpha(x)+(1-t)\alpha(y)$, for all
$x,y\in P$ with $t\in[0,1]$.
\end{definition}
Let $a, b, r>0$ be constants with $P$ and $\alpha$ as defined
above, we note
$$
P_r=\{y\in P: \|y\|b\}\neq\emptyset$ and $\alpha(Ay)>b$, \ for all $y\in
P(\alpha, \ b,\ d) $;
\begin{itemize}
\item[(ii)] $\|Ay\|b$ for all $y\in P(\alpha, b,c)$ with $\|Ay\|>d$.
\end{itemize}
Then $A$ has at least three fixed points $y_1, y_2,y_3$ satisfying
$$
\|y_1\|a,\quad \alpha(y_3)z$ for $t\in(0,1)$, where
$g:[0,1]\times R \to [0, +\infty)$ is defined by
$$
g(t,y)=\begin{cases}
f(t,y)+M, & (t,y)\in[0,1]\times[0,+\infty), \\
f(t,0)+M, & (t,y)\in[0,1]\times(-\infty,0).
\end{cases}
$$
For $v\in P$, define the operator
\begin{align*}
Tv(t)&=-\int_0^{t}\Big(\frac{1}{p(s)}\int_0^{s}p(\tau)g(\tau,v(\tau)
-z(\tau))d\tau\Big)ds\\
&\quad +\frac{1}{1-\alpha}\int_0^1
\Big(\frac{1}{p(s)}\int_0^{s}p(\tau)g(\tau,v(\tau)-z(\tau))d\tau\Big)ds\\
&\quad -\frac{\alpha}{1-\alpha}\int_0^{\eta}
\Big(\frac{1}{p(s)}\int_0^{s}p(\tau)g(\tau,v(\tau)-z(\tau))d\tau\Big)ds,
\end{align*}
where $p$ is defined in Lemma \ref{lem2.1}.
By Lemmas \ref{lem2.1}, \ref{lem2.2} and \ref{lem2.3}, we can check
$T(P)\subseteq P$. It is easy to check $T$ is completely continuous
by Arzela-Ascoli theorem.
In the following, we show that all the conditions of Theorem \ref{thm2.1}
are satisfied. Firstly, we define the nonnegative, continuous
concave functional $\alpha: P\to[0,\infty) $ by
$$
\alpha(v)=\min_{t\in [0,1]}v(t),
$$
Obviously, for every $v\in P$,
$\alpha(v)\leq \|v\|$.
We first show
that $T(\overline{P}_c)\subseteq\overline{P}_c$. Let
$v\in \overline{P}_c$ and $t\in[0,1]$ be arbitrary.
When $v(t)\geq z(t)$, we have $0\leq v(t)-z(t)\leq v(t)\leq c$ and thus
$g(t,v(t)-z(t))=f(t,v(t)-z(t))+M\geq0$. By (A3) we have
$$
g(t,v(t)-z(t))\leq\frac{c}{h}.
$$
When $v(t)b$. That is
$\{v\in P(\alpha, b, \frac{b}{\gamma^2}): \alpha(v)>b\}\neq\emptyset$.
Moreover, if $v\in P(\alpha, b, \frac{b}{\gamma^2})$, then
$\alpha(v)\geq b$, so $b\leq \|v\|\leq \frac{b}{\gamma^2}$.
Thus, $0b.
\end{align*}
Therefore, condition (i) of Theorem \ref{thm2.1} is satisfied with $d=b/\gamma^2$.
Finally, we address condition (iii) of Theorem \ref{thm2.1}. For this we
choose $v\in P(\alpha, b, c)$ with
$\|Tv\|>b/\gamma^2$. Then from above-proved inclusion $T(P)\subseteq P$, we have
$$
\alpha(Tv)=\min_{t\in[0,1]}Tv(t)\geq\gamma\|Tv\|\geq\frac{b}{\gamma}>b.
$$
Hence, condition (iii) of Theorem \ref{thm2.1} holds with $\|Tv\|>b/\gamma^2$.
To sum up, all the hypotheses of Theorem \ref{thm2.1} are satisfied. Hence
$T$ has at least three positive fixed points $v_1$, $v_2$ and
$v_3$ such that
$$
\|v_1\|e,\quad \alpha(v_3)\gamma b>\gamma M\Gamma\geq z(t),
\quad t\in [0,1],\\
v_3(t)\geq \gamma \|v_3\|>\gamma e>\gamma M\Gamma\geq z(t), \quad t\in [0,1].
\end{gather*}
So $u_2= v_2-z$, $u_3=v_3-z$ are two positive solutions of
\eqref{e1.1}. This completes the proof.
\end{proof}
\begin{theorem} \label{thm3.2}
Suppose {\rm (H1)--(H3)} hold, and there exist positive constants
$a_i, b_i, N $ with
$M\Gamma