\documentclass[reqno]{amsart} \usepackage{amssymb} \usepackage{hyperref} \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 $u''(t)+\lambda f(t,u(t))=0$, $00$ 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 00$is a parameter. Throughout, we assume that there exists a constant$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$\Omega _c=\{u\in K:\ \parallel u\parallel w(t)$,$0w(t)$,$01$such that $$T_{\lambda}u\nleqslant u\quad \text{for any } u\in\partial\Omega_{\sigma}.\label{e10}$$ 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 $$u_{n}(t)-w(t)\geq\gamma\| u_{n}\|-\| w\|\to+\infty\ \quad (n\to +\infty).\label{e11}$$ 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 $$\lim_{n\to +\infty }\min_{0\leq t\leq \eta }\frac{\overline{g} (t,u_n(t)-w(t))}{u_n(t)-w(t)}=+\infty .\label{e12}$$ So, there exists a positive integer$N$such that for any$n\geq N$, $$\min_{0\leq t\leq \eta }[u_n(t)-w(t)]\geq \frac \gamma 2\| u_n\| \label{e13}$$ and $$\min_{0\leq t\leq \eta }\frac{\overline{g}(t,u_n(t)-w(t))}{u_n(t)-w(t)}\geq \frac{4\xi }{\lambda \gamma }\Big[ (1-\eta )\int_0^\eta tdt\Big] ^{-1}.\label{e14}$$ 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, $$\| u_{n}\| =\max_{0\leq t\leq 1}u_n(t)\geq \max_{0\leq t\leq 1}(T_\lambda u_n)(t)\geq (T_\lambda u_n)(\eta ).\label{e15}$$ 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. 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