\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations. {\em Electronic Journal of Differential Equations}, Conf. 19 (2010), pp. 151--159.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{151} \title[\hfilneg EJDE-2010/Conf/19/\hfil Third order three-point BVP] {Positive solutions to a nonlinear third order three-point boundary value problem} \author[J. R. Graef, L. Kong, B. Yang\hfil EJDE/Conf/19 \hfilneg] {John R. Graef, Lingju Kong, Bo Yang} % in alphabetical order \address{John R. Graef \newline Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA} \email{John-Graef@utc.edu} \address{Lingju Kong \newline Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA} \email{Lingju-Kong@utc.edu} \address{Bo Yang\newline Department of Mathematics and Statistics, Kennesaw State University, Kennesaw, GA 30144, USA} \email{byang@kennesaw.edu} \thanks{Published September 25, 2010.} \subjclass[2000]{34B15} \keywords{Boundary value problems; existence of positive solutions; \hfill\break\indent nonexistence of positive solutions; nonlinear equations; third order problem; \hfill\break\indent three point boundary conditions} \begin{abstract} We consider a third order three point boundary value problem. Some upper and lower estimates for positive solutions of the problem are proved. Sufficient conditions for the existence and nonexistence of positive solutions for the problem are obtained. An example is included to illustrate the results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \section{Introduction} Recently, second and higher order multi-point boundary value problems have attracted a lot of attention. In 2004, Henderson \cite{H} considered the second order three point boundary value problem $$\label{ee00} \begin{gathered} u''(t)+f(u(t))=0,\quad 0\le t\le 1, \\ u(0)=u(p)-u(1)=0. \end{gathered}$$ In 2006, Graef and Yang \cite{GY-2} studied the third order nonlocal boundary value problem \begin{gather} u'''(t) = g(t)f(u(t)), \quad 0\le t\le 1, \label{ee1} \\ u(0) = u'(p) = u''(1) = 0. \label{ee2} \end{gather} For some other results on third order boundary value problems we refer the reader to the papers \cite{AD,GY,GSZ,HT,M,W}. % Motivated by these works, in this paper we consider the third order three point nonlinear boundary value problem \begin{gather} u'''(t) = g(t)f(u(t)), \quad 0\le t\le 1, \label{bhu}\\ u(0)= u(p) -u(1) = u''(1) = 0. \label{rfv} \end{gather} % To our knowledge, the problem \eqref{bhu}--\eqref{rfv} has not been considered before. The boundary conditions \eqref{rfv} are closely related to some other boundary conditions. Firstly, \eqref{rfv} contains \eqref{ee00} as a part. We also note that $u(p)-u(1)=0$ implies that there exists $\beta\in(p,1)$ such that $u'(\beta)=0$, and therefore the boundary conditions \eqref{rfv} imply $u(0)=u'(\beta)=u''(1)=0.$ Hence, boundary conditions \eqref{rfv} are closely related to the conditions \eqref{ee2}. If we let $p\to 1^-$, then \eqref{rfv} tends to'' $$u(0) = u'(1) = u''(1) = 0, \label{rfvvv}$$ which are often referred to as the (1,2) focal boundary conditions. In this paper, we are interested in the existence and nonexistence of positive solutions of the problem \eqref{bhu}--\eqref{rfv}. % By a {\it positive solution}, we mean a solution $u(t)$ to the boundary value problem such that $u(t)>0$ for $00$ such that $(F_0+\epsilon)B(1+p)^2/4p <1$. Then there exists $H_1 >0$ such that $$f(x)\leq (F_0+\epsilon)x \ \ {\rm for}\ \ 00 such that$$ (f_\infty-\delta)\int_c^{1-c} G(2p/(1+p),s) g(s) a(s)\,ds >1. $$Now, there exists  H_3 >0  such that f(x)\geq (f_\infty-\delta)x for x\geq H_3. Let H_2=H_1 + {H_3}/{c}. If u\in P with \|u\| = H_2, then for c\le t\le 1-c, we have$$ u(t)\geq\min\{t,1-t\}\|u\| \ge c H_2\ge H_3. So, if u\in P with \|u\| = H_2, then % \begin{align*} (Tu)(2p/(1+p)) & \geq \int_c^{1-c} G(2p/(1+p),s) g(s) f(u(s))\,ds \\ & \geq \int_c^{1-c} G(2p/(1+p),s) g(s) (f_\infty-\delta)u(s)ds \\ & \ge (f_\infty-\delta) \|u\| \int_c^{1-c} G(2p/(1+p),s) g(s) a(s)\,ds \\ & > \|u\| \\ &\ge u(2p/(1+p)), \end{align*} % which means Tu\not\le u. So, if we let \Omega_2=\{ u\in X: \|u\|0 such that A(f_0-\epsilon)>1. There exists H_1 >0 such that f(x)\ge (f_0-\epsilon)x for x\geq H_1. If u\in P with \|u\| = H_1, then % \begin{align*} (Tu)(2p/(1+p)) & \ge \int_0^1 G(2p/(1+p),s) g(s) f(u(s))\,ds \\ & \ge \int_0^1 G(2p/(1+p),s) g(s) (f_0-\epsilon)u(s)ds \\ & \ge (f_0-\epsilon) \|u\| \int_0^1 G(2p/(1+p),s) g(s) a(s)\,ds \\ & > \|u\| \\ &\ge u(2p/(1+p)), \end{align*} % which means Tu\not\le u. So, if we let \Omega_1=\{ u\in X\,|\ \|u\|0 such that f(x)\le (F_{\infty}+\delta) x for x\geq H_3. If we let M = \max_{0\leq x\leq H_3}f(x), then f(x)\le M + (F_{\infty}+\delta)x \text{for} x\ge 0. Let $K = M \int_0^1 G(2p/(1+p),s)g(s)ds+1,$ and let H_2 = H_1+K \big(\frac{4p}{(1+p)^2}-(F_{\infty}+\delta)B \big)^{-1}. % Now for each u\in P with \|u\| = H_2, we have \begin{align*} (Tu)(2p/(1+p)) & = \int_0^1 G(2p/(1+p),s) g(s) f(u(s))\,ds \\ & \le \int_0^1 G(2p/(1+p),s) g(s) (M+(F_{\infty}+\delta)u(s))\,ds \\ & < K + (F_{\infty}+\delta)\int_0^{1} G(2p/(1+p),s) g(s) u(s)\,ds \\ & \le K + (F_{\infty}+\delta)\|u\| \int_0^{1} G(2p/(1+p),s) g(s) \,ds \\ &\le K + (F_{\infty}+\delta)B\|u\| \\ & \le \Big(\frac{4p}{(1+p)^2} - (F_{\infty}+\delta)B\Big) H_2 + (F_{\infty}+\delta)B H_2 \\ &= \frac{4p}{(1+p)^2} \|u\| \\ &\le u(2p/(1+p)), \end{align*} % which means  Tu\not\ge u . % So, if we let \Omega_2=\{ u\in X\ |\ \|u\|0 for 0x for all x\in(0,+\infty), then the problem \eqref{bhu}--\eqref{rfv} has no positive solutions. \end{theorem} We conclude the paper with an example. \begin{example} \rm Consider the third-order boundary-value problem \begin{gather} u'''(t)=g(t)f(u(t)), \quad 00 is a parameter. We easily see that F_0=f_0=\lambda and F_{\infty}=f_{\infty}=3\lambda. Calculations indicate that A=\frac{5268393409}{216850636800},\quad B=\frac{33611}{1229312}. $$From Theorem \ref{tt6} we see that if$$ 13.7203 \approx\frac{ 1}{ 3A } < \lambda < \frac{48}{49B}\approx 35.8282, $$then problem \eqref{rfv2}--\eqref{rfv3} has at least one positive solution. From Theorems \ref{tt8} and \ref{tt9}, we see that if$$ \lambda<\frac{ 16 }{49B}\approx 11.9427 \quad\text{or}\quad \lambda>\frac{1}{A}\approx 41.1607,  then problem \eqref{rfv2}--\eqref{rfv3} has no positive solutions. This example shows that our existence and nonexistence conditions work very well. \end{example} \begin{thebibliography}{0} \bibitem{AD} D. R. Anderson and J. M. Davis; \emph{Multiple solutions and eigenvalues for third-order right focal boundary value problems}, J. Math. Anal. Appl. 267 (2002), 135--157. \bibitem{GY} J. R. Graef and B. 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Ma \emph{Multiplicity results for a third order boundary value problem at resonance}, Nonlinear Anal. 32 (1998), 493--499. \bibitem{W} P. J. Y. Wong \emph{Triple fixed-sign solutions for a system of third-order generalized right focal boundary value problems}, Differential \& Difference Equations and Applications, 1139--1148, Hindawi Publ. Corp., New York, 2006. \end{thebibliography} \end{document}