\documentclass[reqno]{amsart} \usepackage[notref,notcite]{showkeys} %\usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 112, 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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/112\hfil Multiple positive solutions] {Multiple Positive solutions for nonlinear third-order three-point boundary-value problems} \author[L.-J. Guo, J.-P. Sun and Y.-H. Zhao\hfil EJDE-2007/112\hfilneg] {Li-Jun Guo, Jian-Ping Sun, Ya-Hong Zhao} % in alphabetical order \address{Department of Applied Mathematics \\ Lanzhou University of Technology \\ Lanzhou, Gansu, 730050, China} \email[L.-J. Guo]{school520@lut.cn} \email[J.-P. Sun (Corresponding author) ]{jpsun@lut.cn} \email[Y.-H. Zhao]{zhaoyahong88@sina.com} \thanks{Submitted April 18, 2007. Published August 18, 2007.} \thanks{Supported by the NSF of Gansu Province of China} \subjclass[2000]{34B10, 34B18} \keywords{Third-order boundary value problem; positive solution; \hfill\break\indent three-point boundary value problem; existence; cone; fixed point} \begin{abstract} This paper concerns the nonlinear third-order three-point bound\-ary-value 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 $1<\alpha <\frac 1\eta$. First, we establish the existence of at least three positive solutions by using the well-known Leggett-Williams fixed point theorem. And then, we prove the existence of at least $2m-1$ positive solutions for arbitrary positive integer $m$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} Third-order differential equations arise in a variety of different 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{g1}. Recently, third-order boundary value problems (BVPs for short) have received much attention. For example, \cite{d1,f1,h1,l2,y2} discussed some third-order two-point BVPs, while \cite{a1,a2,m1,s1,y1} studied some third-order three-point BVPs. In particular, Anderson \cite{a1} obtained some existence results of positive solutions for the BVP \begin{gather} x'''(t)=f(t,x(t)),\quad t_1\leq t\leq t_3, \label{0.1}\\ x(t_1)=x'(t_2)=0,\quad \gamma x(t_3)+\delta x''(t_3)=0 \label{0.2} \end{gather} by using the well-known Guo-Krasnoselskii fixed point theorem \cite{g2,k1} and Leggett-Williams fixed point theorem \cite{l1}. In 2005, the author in \cite{s1} established various results on the existence of single and multiple positive solutions to some third-order differential equations satisfying the following three-point boundary conditions $$x(0)=x'(\eta )=x''(1)=0, \label{0.3}$$ where $\eta \in [\frac 12,1)$. The main tool in \cite{s1} was the Guo-Krasnoselskii fixed point theorem. Recently, motivated by the above-mentioned excellent works, we \cite{g3} considered the third-order three-point BVP \begin{gather} u'''(t)+h(t)f(u(t))=0,\quad t\in (0,1), \label{1.1}\\ u(0)=u'(0)=0,\quad u'(1)=\alpha u'(\eta ), \label{1.2} \end{gather} where $0<\eta <1$. By using the Guo-Krasnoselskii fixed point theorem, we obtained the existence of at least one positive solution for the BVP \eqref{1.1}--\eqref{1.2} under the assumption that $1<\alpha <\frac 1\eta$ and $f$ is either superlinear or sublinear. In this paper, we will continue to study the BVP \eqref{1.1}--\eqref{1.2}. First, some existence criteria for at least three positive solutions to the BVP \eqref{1.1}--\eqref{1.2} are established by using the well-known Leggett-Williams fixed point theorem. And then, for arbitrary positive integer $m$, existence results for at least $2m-1$ positive solutions are obtained. In the remainder of this section, we state some fundamental concepts and the Leggett-Williams fixed point theorem. Let $E$ be a real Banach space with cone $P$. A map $\sigma :P\to[0,+\infty )$ is said to be a nonnegative continuous concave functional on $P$ if $\sigma$ is continuous and $\sigma (tx+(1-t)y)\geq t\sigma (x)+(1-t)\sigma (y)$ for all $x$, $y\in P$ and $t\in [0,1]$. Let $a$, $b$ be two numbers such that $0a\} \neq \emptyset$ 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, \quad \sigma (x_3)\frac{d_1}C,\quad u\in [ d_1,\frac{d_1}\gamma ] , \label{2} \\ f(u)<\frac cD,\quad u\in [ 0,c] . \label{2.1} \end{gather} Then the BVP \eqref{1.1}--\eqref{1.2} has at least three positive solutions. \end{theorem} \begin{proof} Let the Banach space E=C[ 0,1]  be equipped with the norm \[ \| u\| =\max_{0\leq t\leq 1}| u(t)| .$ We denote $P=\{ u\in E: u(t)\geq 0,\; t\in [ 0,1] \}.$ Then, it is obvious that $P$ is a cone in $E$. For $u\in P$, we define $\sigma (u)=\min_{t\in [ \frac \eta \alpha ,\eta ] }u(t)$ and $$Au(t)=\int_0^1G(t,s)h(s)f(u(s))ds,\quad t\in [0,1]. \label{3.1}$$ It is easy to check that $\sigma$ is a nonnegative continuous concave functional on $P$ with $\sigma (u)\leq \| u\|$ for $u\in P$ and that $A:P\to P$ is completely continuous and fixed points of $A$ are solutions of the BVP \eqref{1.1}--\eqref{1.2}. We first assert that if there exists a positive number $r$ such that $f(u)<\frac rD$ for $u\in [ 0,r]$, then $A:\overline{P_r} \to P_r$. Indeed, if $u\in \overline{P_r}$, then for $t\in [0,1]$, \begin{align*} (Au)(t)&=\int_0^1G(t,s)h(s)f(u(s))ds\\ &<\frac rD\int_0^1G(t,s)h(s)ds\\ &\leq \frac rD\max_{t\in [0,1]}\int_0^1G(t,s)h(s)ds=r. \end{align*} Thus, $\| Au\| d_1\} \neq \emptyset$ and $\sigma (Au)>d_1$ for all $u\in P(\sigma ,d_1,d_1/\gamma )$. In fact, the constant function $\frac{d_1+d_1/\gamma }2\in \{ u\in P(\sigma ,d_1,d_1/\gamma ):\sigma (u)>d_1\} .$ Moreover, for $u\in P(\sigma ,d_1,d_1/\gamma )$, we have $d_1/\gamma \geq \| u\| \geq u(t)\geq \min_{t\in [ \frac \eta \alpha ,\eta ] }u(t)=\sigma (u)\geq d_1$ for all $t\in [ \frac \eta \alpha ,\eta ]$. Thus, in view of (\ref{2}), we see that \begin{align*} \sigma (Au) &=\min_{t\in [ \frac \eta \alpha ,\eta ] }\int_0^1G(t,s)h(s)f(u(s))ds\\ &\geq \min_{t\in [ \frac \eta \alpha ,\eta ] }\int_{\frac \eta \alpha }^\eta G(t,s)h(s)f(u(s))ds \\ &> \frac{d_1}C\min_{t\in [ \frac \eta \alpha ,\eta ] }\int_{\frac \eta \alpha }^\eta G(t,s)h(s)ds=d_1 \end{align*} as required. Finally, we assert that if $u\in P(\sigma ,d_1,c)$ and $\|Au\| >d_1/\gamma$, then $\sigma (Au)>d_1$. To see this, we suppose that $u\in P(\sigma ,d_1,c)$ and $\| Au\| >d_1/\gamma$, then, by Lemma \ref{lem2.2} and Lemma \ref{lem2.3}, we have \begin{align*} \sigma (Au)&=\min_{t\in [ \frac \eta \alpha ,\eta ] }\int_0^1G(t,s)h(s)f(u(s))ds\\ &\geq \gamma \int_0^1g(s)h(s)f(u(s))ds\geq \gamma \int_0^1G(t,s)h(s)f(u(s))ds \end{align*} for all $t\in [ 0,1]$. Thus $\sigma (Au)\geq \gamma \max_{t\in [ 0,1] }\int_0^1G(t,s)h(s)f(u(s))ds=\gamma \| Au\| >\gamma \frac{d_1} \gamma =d_1.$ To sum up, all the hypotheses of the Leggett-Williams theorem are satisfied. Hence $A$ has at least three fixed points, that is, the BVP \eqref{1.1}--\eqref{1.2} has at least three positive solutions $u$, $v$, and $w$ such that \[ \| u\| d_0, \quad \min_{t\in [ \frac \eta \alpha ,\eta ] }w(t)\frac{a_j}C,\quad u\in [ a_j,\frac{a_j}\gamma ] ,\quad 1\leq j\leq m-1. \label{4.2} \end{gather} Then, the BVP \eqref{1.1}--\eqref{1.2} 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 (\ref{4.1}) that $A:\overline{P_{d_1}} \to P_{d_1}$, then, it follows from Schauder fixed point theorem that the BVP \eqref{1.1}--\eqref{1.2} has at least one positive solution in $\overline{P_{d_1}}$. Next, we assume that this conclusion holds for $m=k$. In order to prove 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}C,\quad u\in [ a_j,\frac{a_j}\gamma ] ,\; 1\leq j\leq k. \label{6} \end{gather} By assumption, the BVP \eqref{1.1}--\eqref{1.2} 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}, (\ref{5}) and (\ref{6}) that the BVP \eqref{1.1}--\eqref{1.2} has at least three positive solutions$u$,$v$, and$w$in$\overline{P_{d_{k+1}}}\$ such that \[ \| u\| d_k, \quad \min_{t\in [ \frac \eta \alpha ,\eta ] }w(t)