\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 118, 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-2012/118\hfil Multiple positive solutions] {Multiple positive solutions for a third-order three-point BVP with sign-changing Green's function} \author[J.-P. Sun, J. Zhao\hfil EJDE-2012/118\hfilneg] {Jian-Ping Sun, Juan Zhao} % in alphabetical order \address{Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China} \email[Jian-Ping]{jpsun@lut.cn} \email[Juan Zhao]{z\_1111z@163.com} \thanks{Submitted May 29, 2012. Published July 14, 2012.} \subjclass[2000]{34B10, 34B18} \keywords{Third-order three-point boundary-value problem; \hfill\break\indent sign-changing Green's function; positive solution} \begin{abstract} This article concerns the third-order three-point boundary-value problem \begin{gather*} u'''(t)=f(t,u(t)),\quad t\in [0,1], \\ u'(0)=u(1)=u''(\eta)=0. \end{gather*} Although the corresponding Green's function is sign-changing, we still obtain the existence of at least $2m-1$ positive solutions for arbitrary positive integer $m$ under suitable conditions on $f$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} Third-order differential equations arise from a variety of areas of applied mathematics and physics, e.g., in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity driven flows and so on \cite{4}. Recently, the existence of single or multiple positive solutions to some third-order three-point boundary-value problems (BVPs for short) has received much attention from many authors. For example, in 1998, by using the Leggett-Williams fixed point theorem, Anderson \cite{0} proved the existence of at least three positive solutions to the problem \begin{gather*} -x'''(t)+f(x(t))=0,\quad t\in [0,1],\\ x(0)=x'(t_2)=x''(1)=0, \end{gather*} where $t_2\in [\frac{1}{2},1)$. In 2003, Anderson \cite{1.1} obtained some existence results of positive solutions for the problem \begin{gather*} x'''(t)=f(t,x(t)),\quad t_1\leq t\leq t_3,\\ x(t_1)=x'(t_2)=0,\quad \gamma x(t_3)+\delta x''(t_3)=0. \end{gather*} The main tools used were the Guo-Krasnosel'skii and Leggett-Williams fixed point theorems. In 2005, Sun \cite{9} studied the existence of single and multiple positive solutions for the singular BVP \begin{gather*} u'''(t)-\lambda a(t)F(t,u(t))=0,\quad t\in (0,1),\\ u(0)=u'(\eta)=u''(1)=0, \end{gather*} where $\eta\in [\frac{1}{2},1)$, $\lambda$ was a positive parameter and $a(t)$ was a nonnegative continuous function defined on $(0, 1)$. His main tool was the Guo-Krasnosel'skii fixed point theorem. In 2008, by using the Guo-Krasnosel'skii fixed point theorem, Guo, Sun and Zhao \cite{3} obtained the existence of at least one positive solution for the problem \begin{gather*} u'''(t)+h(t)f(u(t))=0,\quad t\in (0,1),\\ u(0)=u'(0)=0,\quad u'(1)=\alpha u'(\eta ), \end{gather*} where $0<\eta <1$ and $1<\alpha <1/\eta$. For more results concerning the existence of positive solutions to third-order three-point BVPs, one can refer to \cite{3.1,3.2,13,12,10,11}. It is necessary to point out that all the above-mentioned works are achieved when the corresponding Green's functions are positive, which is a very important condition. A natural question is that whether we can obtain the existence of positive solutions to some third-order three-point BVPs when the corresponding Green's functions are sign-changing. It is worth mentioning that Palamides and Smyrlis \cite{7} discussed the existence of at least one positive solution to the singular third-order three-point BVP with an indefinitely signed Green's function \begin{gather*} u'''(t)=a(t)f(t,u(t)),\quad t\in (0,1),\\ u(0)=u(1)=u''(\eta)=0,\quad \eta\in (\frac{17}{24},1). \end{gather*} Their technique was a combination of the Guo-Krasnosel'skii fixed point theorem and properties of the corresponding vector field. The following equality $$\max_{t\in[0,1]}\int_0^1G(t,s)a(s)f(s,u(s))ds =\int_0^1\max_{t\in[0,1]} G( t,s)a(s)f(s,u(s))ds \label{0}$$ played an important role in the process of their proof. Unfortunately, the equality \eqref{0} is not right. For a counterexample, one can refer to our paper \cite{16}. Motivated greatly by the above-mentioned works, in this paper we study the following third-order three-point BVP $$\begin{gathered} u'''(t)=f(t,u(t)),\quad t\in [0,1], \\ u'(0)=u(1)=u''(\eta)=0, \end{gathered}\label{1.1}$$ where $f\in C([0,1]\times[0,+\infty),\ [0,+\infty))$ and $\eta\in(\frac{1}{2},1)$. Although the corresponding Green's function is sign-changing, we still obtain the existence of at least $2m-1$ positive solutions for arbitrary positive integer $m$ under suitable conditions on $f$. In the remainder of this section, we state some fundamental concepts and the Leggett-Williams fixed point theorem \cite{2}. Let $E$ be a real Banach space with cone $P$. A map $\sigma: P \to (-\infty,+\infty)$ is said to be a concave functional if $\sigma(tx+(1-t)y)\geq t\sigma(x)+(1-t)\sigma(y)$ for all $x,y\in P$ and $t\in [0,1]$. Let $a$ and $b$ be two numbers with $0a\}\neq \emptyset$ and $\sigma(Ax)>a$ for $x\in P(\sigma,a,b)$; \item[(ii)] $\| Ax\|a$ for $x\in P(\sigma,a,c)$ with $\| Ax\|>b$. \end{itemize} Then $A$ has at least three fixed points $x_1,x_2,x_{3}$ in $\overline{P_c}$ satisfying \| x_1\|d,\ \sigma(x_{3})\frac{a}{H_2},\quad t\in [1-\theta,\theta],\; u\in[a,\frac{a}{\theta^{*}}],\label{3.2} \\ f(t,u)<\frac{c}{H_1},\quad t\in [0,\eta],\; u\in[0,c]. \label{3.3} \end{gather} Then \eqref{1.1} has at least three positive solutions u, v and w satisfying \[ \|u\| a\}\neq \emptyset and \sigma(Au) > a for all u \in P(\sigma, a, \frac{a}{\theta^{*}}). In fact, the constant function \frac{a+\frac{a}{\theta^{*}}}{2} belongs to \{u \in P(\sigma,a,\frac{a}{\theta^{*}}): \sigma(u) > a\}. On the one hand, for u\in P(\sigma,a,\frac{a}{\theta^{*}}), we have $$a\leq\sigma(u)=\min_{t\in[1-\theta,\theta]}u(t)\leq u(t) \leq \|u\|\leq \frac{a}{\theta^{*}} \label{3.35}$$ for all t\in [1-\theta,\theta]. Also, for any u\in P and t\in [1-\theta,\theta], we have \begin{align*} &\int_0^{1-\theta} G(t,s)f(s,u(s))ds+\int_\theta^\eta G(t,s)f(s,u(s))ds +\int_\eta^1 G(t,s)f(s,u(s))ds\\ &\geq \int_0^{1-\theta} (1-t)s f(s,u(s))ds -\int_\eta^1 \frac{(1-s)^{2}}{2}f(s,u(s))ds\\ &\geq f(\eta,u(\eta))[\int_0^{1-\theta} (1-t)s ds -\int_\eta^1 \frac{(1-s)^{2}}{2}ds]\\ &\geq f(\eta,u(\eta))[\int_0^{1-\theta} (1-t)s ds -\int_\theta^1 \frac{(1-s)^{2}}{2}ds]\\ &= f(\eta,u(\eta))[\frac{(1-t)(1-\theta)^{2}}{2}-\frac{(1-\theta)^{3}}{6}]\\ &\geq f(\eta,u(\eta))[\frac{(1-\theta)(1-\theta)^{2}}{2} -\frac{(1-\theta)^{3}}{6}]\\ &= f(\eta,u(\eta))\frac{(1-\theta)^{3}}{3} \geq 0, \end{align*} which together with \eqref{3.2} and \eqref{3.35} implies \begin{align*} \sigma(Au) &= \min_{t\in[1-\theta,\theta]}\int_0^1 G(t,s)f(s,u(s))ds\\ &\geq \min_{t\in[1-\theta,\theta]}\int_{1-\theta}^\theta G(t,s)f(s,u(s))ds\\ &> \frac{a}{H_2}\min_{t\in[1-\theta,\theta]}\int_{1-\theta}^\theta G(t,s)ds = a \end{align*} for u\in P(\sigma,a,\frac{a}{\theta^{*}}). Finally, we verify that if u \in P(\sigma, a, c) and \| Au\| > a/\theta^{*} , then \sigma(Au) > a. To see this, we suppose that u \in P(\sigma, a, c) and \| Au\| > a/\theta^{*}. Then it follows from Au \in P that \[ \sigma(Au)= \min_{t\in[1-\theta,\theta]}(Au)(t) \geq \theta^{*}\| Au\| > a. To sum up, all the hypotheses of the Leggett-Williams fixed point theorem are satisfied. Therefore, $A$ has at least three fixed points; that is, \eqref{1.1} has at least three positive solutions $u, v$ and $w$ satisfying \[ \|u\|\frac{a_j}{H_2},\quad t\in [1-\theta,\theta],\; u\in[a_j,\frac{a_j}{\theta^{*}}],\; 1\leq j \leq m-1.\label{3.5} \end{gather} Then \eqref{1.1} has at least $2m-1$ positive solutions in $\overline{P_{d_{m}}}$. \end{theorem} \begin{proof} We use induction on $m$. First, for $m = 1$, we know from \eqref{3.4} that $A :\overline{P_{d_1}}\to P_{d_1}$. Then it follows from Schauder fixed point theorem that \eqref{1.1} has at least one positive solution in $\overline{P_{d_1}}$. Next, we assume that this conclusion holds for $m = k$. To show that this conclusion also holds for $m=k+1$, we suppose that there exist numbers $d_i$ $(1\leq i\leq k+1)$ and $a_j$ $(1\leq j \leq k)$ with $0\frac{a_j}{H_2},\quad t\in [1-\theta,\theta],\; u\in[a_j,\frac{a_j}{\theta^{*}}],\; 1\leq j \leq k.\label{3.7} \end{gather} By assumption, \eqref{1.1} has at least$2k-1$positive solutions$u_i\ (i = 1,2,\dots,2k-1)$in$\overline{P_{d_k}}$. At the same time, it follows from Theorem \ref{thm3.1}, \eqref{3.6} and \eqref{3.7} that \eqref{1.1} has at least three positive solutions$u, v$and$w$in$\overline{P_{d_{k+1}}}\$ such that \[ \|u\|