\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 154, 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-2007/154\hfil Existence of positive solutions] {Existence of positive solutions for the symmetry three-point boundary-value problem} \author[Q. Ma \hfil EJDE-2007/154\hfilneg] {Qiaozhen Ma} \address{Qiaozhen Ma \newline College of Mathematics and Information Science, Northwest Normal University,\newline Lanzhou, Gansu, 730070, China} \email{maqzh@nwnu.edu.cn} \thanks{Submitted November 24, 2006. Published November 16, 2007.} \thanks{Supported by grants 10671158 from NSFC, 10626042 from the Mathematical Tianyuan \hfill\break\indent Foundation, and 3ZS061-A25-016 from the Natural Sciences Foundation of Gansu, and \hfill\break\indent NWNU-KJCXGC-03-40} \subjclass[2000]{34B10} \keywords{Positive solution; three-point boundary value problem} \begin{abstract} In this paper, we show the existence of single and multiple positive solutions for the symmetry three-point boundary value problem under suitable conditions by using classical 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} Since Gupta \cite{g2} studied three-point boundary value problems for the nonlinear ordinary differential equation, many classical results have been obtained by using Leray-Schauder continuation theorem, nonlinear alternatives of Leray-Schauder and coincidence degree theory. For more information, we refer the reader to \cite{f1,g2,m1,m2} and reference therein. The study of multi-point boundary-value problems for linear second-order differential equations was initiated by II'in and Moiseev \cite{i1}. While the multi-point boundary value problem arise in the different areas of applied mathematics and physics. For instance, many problems in the theory of elastic stability can be handled as a multi-point problem \cite{t1}. Therefore, it's necessary to continue to extend and investigate. Ma \cite{m1}, by using fixed-point index theorems and Leray-Schauder degree and upper and lower solutions, considered the multiplicity of positive solutions of the problem \begin{gather} u''+\lambda h(t)f(u)=0, \quad t \in (0, 1), \label{e1.1}\\ u(0)=0, \quad u(1)=\alpha u(\eta), \label{e1.2} \end{gather} where $0<\eta<1$, $0<\alpha<1/\eta$, assuming that $f\in C([0,\infty), [0,\infty))$, $h\in C([0,1), [0,\infty))$, and $f$ is superlinear. In the present paper, we study the existence of single and multiple positive solutions to nonlinear symmetry three-point boundary value problem \begin{gather} u''+\lambda a(t)f(u)=0, \quad t \in (0, 1), \label{e1.3} \\ u(0)=\beta u(\eta), \quad u(1)=\alpha u(\eta). \label{e1.4} \end{gather} Where $\lambda>0$ is a positive parameter, $\alpha>0$, $\beta>0$, $0<\eta<1$. Clearly, problem \eqref{e1.3}-\eqref{e1.4} is more generic than \eqref{e1.1}-\eqref{e1.2}, that is to say, our problem is \eqref{e1.1}-\eqref{e1.2} for $\beta=0$. Moreover, \eqref{e1.3}-\eqref{e1.4} is transformed immediately into the classical Dirichlet problem for $\alpha=\beta=0$. And when $\beta=0$, $\alpha=1$, $\eta\to 1$ problem \eqref{e1.3}-\eqref{e1.4} is changed into the mixed boundary value problem. In addition, our results will be obtained under conditions that do not require $f$ to be either superlinear or sublinear. In short, our problem gives a frame to these problems under more generic conditions. We make the following assumptions. \begin{itemize} \item[(i)] $a\in C([0,1], [0,+\infty))$ and there exists $x_0 \in [0,1]$ such that $a(x_0)>0$. \item[(ii)] $f \in C([0,+\infty), [0,+\infty))$ and there exist nonnegative constants in the extended reals, $f_0$, $f_\infty$, such that $$f_0=\lim_{u\to 0+}\frac{f(u)}{u}, \quad f_\infty=\lim_{u\to \infty}\frac{f(u)}{u}.$$ \item[(iii)] $f(0)>0$, for $t \in[0,1]$. \end{itemize} \begin{remark} \label{rmk1.1} \rm It is easy to see that if (iii) holds, then there exist two constants $a, b \in (0, \infty)$, such that $00$ such that $$\frac{1}{\gamma B(f_\infty -\varepsilon)}\leq \lambda\leq\frac{1}{A(f_0+\varepsilon)}. \label{e3.1}$$ By the definition of $f_0$, there exists $H_1>0$ such that $f(x)\leq(f_0+\varepsilon)x$ for $x\in[0,H_1]$. Let $u\in K$ with $\|u\|=H_1$, by \eqref{e2.3} and \eqref{e3.1}, we conclude that \begin{aligned} A_\lambda u(t) &\leq \frac{\lambda\beta t}{(1-\alpha\eta)-\beta(1-\eta)} \int_{0}^{\eta}(\eta-s)a(s)f(u(s))ds\\ &\quad +\frac{\lambda(t+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)} \int_{0}^{1}(1-s)a(s)f(u(s))ds\\ &\leq \frac{\lambda\beta }{(1-\alpha\eta)-\beta(1-\eta)} \int_{0}^{1}(1-s)a(s)f(u(s))ds\\ &\quad +\frac{\lambda(1+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)} \int_{0}^{1}(1-s)a(s)f(u(s))ds\\ &=\frac{\lambda(1+\beta+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)} \int_{0}^{1}(1-s)a(s)f(u(s))ds\\ &\leq\frac{\lambda(1+\beta+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)} \int_{0}^{1}(1-s)a(s)(f_0+\varepsilon)u(s)ds\\ &\leq\lambda A(f_0+\varepsilon)\|u\|\leq \|u\|. \end{aligned} \label{e3.2} As a result, $\|A_\lambda u\| \leq \|u\|$. Let $\Omega_1=\{u\in K: \|u\|0$ such that $f(x)\geq (f_\infty-\varepsilon)x$, for every $x\in [\hat{H}_2, \infty)$. Denote $H_2=\max\{2H_1, \frac{\hat{H}_2}{\gamma}\}$, $\Omega_2=\{u\in K: \|u\|0$ such that $$\frac{1}{\gamma B(f_0 -\varepsilon)}\leq \lambda\leq\frac{1}{A(f_\infty+\varepsilon)}. \label{e3.5}$$ By the definition of $f_0$, there exists $H_3>0$ such that $f(x)\geq(f_0-\varepsilon)x$ for $x\in[0,H_3]$. Let $u\in K$ with $\|u\|=H_3$ such that $\min_{t\in[0,1]}u(t)\geq\gamma\|u\|$. Similar to the estimates of \eqref{e3.4}, we obtain \begin{aligned} A_\lambda u(0)&\geq\frac{\lambda\beta(1- \eta)}{(1-\alpha\eta)-\beta(1-\eta)} \int_{0}^{\eta}sa(s)f(u(s))ds\\ &\geq\frac{\lambda\beta(1- \eta)}{(1-\alpha\eta)-\beta(1-\eta)} \int_{0}^{\eta}sa(s)(f_0-\varepsilon)u(s)ds\\ &\geq\lambda\gamma B(f_0-\varepsilon)\|u\| \geq \|u\|. \end{aligned} \label{e3.6} Hence, it follows that $\|A_\lambda u\|\geq \|u\|$. Set $\Omega_1=\{u\in K: \|u\|\max\{2H_3, \gamma^{-1}\tilde{H}_4\}$ such that $f(x)\leq f(H_4)$, for $x\in [0,H_4]$. Let $u\in K$ with $\|u\|=H_4$, we have \begin{aligned} A_\lambda u(t) &\leq\frac{\lambda(1+\beta+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)} \int_{0}^{1}(1-s)a(s)f(u(s))ds\\ &\leq\frac{\lambda(1+\beta+\beta\eta)}{(1-\alpha\eta)-\beta(1-\eta)} \int_{0}^{1}(1-s)a(s)f(H_4)ds\\ &\leq\lambda A(f_\infty+\varepsilon) H_4 \leq \|u\|. \end{aligned} \label{e3.8} Let \$\Omega_2=\{u\in K: \|u\|