\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 133, 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 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/133\hfil Positive solutions for semi-positone Neumann BVP] {Positive solutions for singular semi-positone Neumann boundary-value problems} \author[Y. P. Sun \& Y. Sun\hfil EJDE-2004/133\hfilneg] {Yong-Ping Sun, Yan Sun} % in alphabetical order \address{Yong-Ping Sun \hfill\break Department of Mathematics, Qufu Normal University \\ Qufu, Shandong 273165, China. \hfill\break Department of Fundamental Courses, Hangzhou Radio \& TV University \\ Hangzhou, Zhejiang 310012, China} \email{syp@mail.hzrtvu.edu.cn} \address{Yan Sun \hfill\break Department of Mathematics, Qufu Normal University \\ Qufu, Shandong 273165, China} \email{ysun@163169.net} \date{} \thanks{Submitted October 12, 2004. Published November 16, 2004.} \thanks{Supported by NSFC (19871075), NSFSP (Z2003A01) and EDZP (20040495)} \subjclass[2000]{34B10, 34B15} \keywords{Positive solution; semi-positone; fixed points; cone; \hfill\break\indent singular Neumann boundary-value problem} \begin{abstract} In this paper, we study the singular semi-positone Neumann bound\-ary-value problem \begin{gather*} -u''+m^2u=\lambda f(t,u)+g(t,u),\quad 00$is a constant,$\lambda>0$is a parameter,$f: (0,1)\times[0,+\infty)\to [0,+\infty)$and$g:[0,1]\times [0,+\infty)\to (-\infty,+\infty)$are continuous. We say problem \eqref{e1.1} is singular because$f$may be singular at$t=0$and/or$t=1$. When$g(t,u)\not\equiv0$, problem \eqref{e1.1} is a semi-positone problem, this situation arises naturally in chemical reaction theory \cite{a}. In recent years, attention has been paid to \eqref{e1.1} in the case of$g(t,u)\equiv 0$; see, for example, \cite{rs,do,ma} and the references therein. Attention has been paid also to the semi-positone boundary-value problem; see, for example, \cite{ag,mwr,x} and the references therein. As far as the authors know, there are no existence results for the singular semi-positone NBVP. Recently, Xu \cite{x} studied the existence of positive solutions for the singular semi-positone boundary-value problem \begin{gather*} u''+f(t,u)+q(t)=0 ,\quad 00$ is a constant. \item[(H3)] $0<\int_0^1G(s,s)p(s)ds<+\infty$. \item[(H4)] $\lim_{u\to+\infty}\frac{f(t,u)}{u}=+\infty$ uniformly on any compact subinterval of $(0,1)$. \end{itemize} Let \begin{gather*} C^+[0,1]=\{u\in C[0,1]:u(t)\geq 0,\ 0\leq t\leq 1\},\\ K=\{u:u\in C^+[0,1],\min_{\theta\leq t\leq 1-\theta}u(t)\geq M_{\theta}\|u\|\}. \end{gather*} It is obvious that $C^+[0,1]$ and $K$ are cones of $E$. Let $v(t)$ be the solution of the boundrary-value problem \begin{gather*} -v''+m^2v=M ,\ 0v(t)$,$00$such that$\|u\|\leq L$for all$u\in D$, hence$[u(s)-x(s)]^*\leq u(s)\leq \|u\|\leq L.We have \begin{align*}& \Big| (T_\lambda u)(t)-(T_nu)(t)\Big|\\ &\leq \lambda \int_0^{1/n}G(t,s)\Big|F(s,u(s))-F(\frac{1}{n},u(s))\Big|ds\\ &\quad +\lambda\int_{1-\frac{1}{n}}^1G(t,s)\Big|F(s,u(s))-F(1-\frac{1}{n},u(s))\Big|ds\\ &\leq 2\lambda\Big(\int_0^{1/n}G(t,s)[p(s)q(u(s))+M]ds+ \int_{1-\frac{1}{n}}^1G(t,s)[p(s)q(u(s))+M]ds\Big)\\ &\leq 2\lambda\max _{0\leq x\leq L}q(x)\Big(\int_0^{1/n}G(s,s)p(s)ds+ \int_{1-\frac{1}{n}}^1G(s,s)p(s)ds\Big)\\ &\quad +2M\Big(\int_0^{1/n}G(s,s)ds+ \int_{1-\frac{1}{n}}^1G(s,s)ds\Big)\\ & \to 0(n\to \infty). \end{align*} Therefore,T_n$converge uniformly to$T_\lambda$on any bounded subset of$E$. This implies that$T_\lambda$is a completely continuous operator. \end{proof} The following Krasnosel'skii fixed point theorem in a cone plays an important role in proving the main result \cite{g}. \begin{theorem}\label{thm4} Let$E$be Banach space and$K\subset E$be a cone in$E$. Suppose\$\Omega_1$\ and\$\Omega_2$are open subset of$E$with$0 \in \Omega_1$and$\overline{\Omega}_1 \subset \Omega_2$. Let$\ T:K\cap (\overline{\Omega}_2\setminus \Omega_1)\to K$be a completely continuous operator such that \begin{itemize} \item[(A)]$\|Tu\|\leq \|u\|$for all$u\in K\cap \partial \Omega_1 $and$ \|Tu\| \geq \|u\|$for all$u\in K\cap \partial \Omega_2$or \item[(B)]$ \|Tu\|\leq \|u\|$for all$u\in K\cap \partial \Omega_2 $and$ \|Tu\|\geq \|u\|$for all$u\in K\cap \partial \Omega_1$. \end{itemize} Then$T$has a fixed point in$K\cap (\overline{\Omega}_2\setminus \Omega_1)$. \end{theorem} \section{Main Result} In this section, we present and prove our main result. \begin{theorem}\label{thm1} Suppose (H1)--(H4)hold, then \eqref{e1.1} has at least one positive solution$u\in C(0,1)\cap C^2[0,1]$if $$0<\lambda\leq \Big[\max_{0\leq t\leq r}q(\tau)\int_0^1G(s,s)p(s)ds)\Big]^{-1},$$ where$r=\max\big\{1+2M\int_0^1G(s,s)ds,\ \frac{M}{\rho C}\big\}$. \end{theorem} \begin{proof} By Lemma \ref{lem3}, we know that$T_\lambda\$ is a completely continuous operator. Let $$\Omega_1=\{u\in C[0,1]:\|u\|0 such that$$ \frac{f(s,u)}{u}>N,\quad\hbox{for }(s,u)\in [\theta,1-\theta]\times [B,\infty). $$Set$$ \Omega_2=\{u\in C[0,1]:\|u\|