\documentclass[twoside]{article} \usepackage{amsfonts,amsmath} \pagestyle{myheadings} \markboth{\hfil Positive and monotone solutions \hfil EJDE--2002/??} {EJDE--2002/??\hfil Panos K. Palamides \hfil} \begin{document} \title{\vspace{-1in} \parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. ??, pp. 1--16. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Positive and monotone solutions of an m-point boundary-value problem % \thanks{ \emph{Mathematics Subject Classifications:} 34B10, 34B18, 34B15. \hfil\break \indent {\em Key words:} multipoint boundary value problems, positive monotone solution, vector field, \hfill\break\indent sublinear, superlinear, Kneser's property, solution's funel. \hfil\break \indent \copyright 2002 Southwest Texas State University. \hfil\break \indent Submitted January 10, 2002. Published February 18, 2002.} } \date{} \author{Panos K. Palamides} \maketitle \begin{abstract} We study the second-order ordinary differential equation $$y''(t)=-f(t,y(t),y'(t)),\quad 0\leq t\leq 1,$$ subject to the multi-point boundary conditions $$\alpha y(0)\pm \beta y'(0)=0,\quad y(1)=\sum_{i=1}^{m-2}\alpha_iy(\xi_i)\,.$$ We prove the existence of a positive solution (and monotone in some cases) under superlinear and/or sublinear growth rate in $f$. Our approach is based on an analysis of the corresponding vector field on the $(y,y')$ face-plane and on Kneser's property for the solution's funnel. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \section{Introduction} Recently an increasing interest has been observed in investigating the existence of positive solutions of boundary-value problems. This interest comes from situations involving nonlinear elliptic problems in annular regions. Erbe and Tang \cite{ET} noted that, if the boundary-value problem \begin{equation*} -\Delta u=F(|x|,u)\quad\text{in }R<|x|<\hat{R} \end{equation*} with \begin{gather*} u=0\quad\mbox{for }|x| =R,\quad u=0 \quad\mbox{for }|x| =\hat {R}; \quad \mbox{or} \\ u=0 \quad\mbox{for }|x| =R,\quad \frac{\partial u}{\partial|x|} =0 \quad \mbox{for } |x| =\hat{R};\quad\mbox{or} \\ \frac{\partial u}{\partial|x|}=0\quad\mbox{for } |x|=R, \quad u=0 \quad \mbox{for }|x|=\hat{R} \end{gather*} is radially symmetric, then the boundary-value problem can be transformed into the scalar Sturm-Liouville problem \begin{gather} x''(t)=-f(t,x(t)),\quad 0\leq t\leq1, \label{E0} \\ \alpha x(0)-\beta x'(0)=0,\quad \gamma x(1)+\delta x'(1)=0. \label{C0} \end{gather} where $\alpha$, $\beta$, $\gamma$, $\delta$ are positive constants. By a positive solution of (\ref{E0})-(\ref{C0}), we mean a function $x(t)$ which is positive for $00$. \end{enumerate} Also nonlinear boundary constraints have been studied, among others by Thompson \cite{TO} and by the author of this paper and Jackson \cite{JP}. There are common ingredients in these papers: an (assumed) Nagumo-type growth condition on the nonlinearity $f$ or/and the presence of upper and lower solutions. The multi-point boundary-value problem for second-order ordinary differential equations was initiated by Ilin and Moiseev \cite{IM-1,IM-2}. Gupta \cite{GU} studied the three-point boundary-value problems for nonlinear ordinary differential equations. Since then, more general nonlinear multi-point boundary-value problems have been studied by several authors. Most of them used the Leray-Schauder continuation theorem, nonlinear alternatives of Leray-Schauder, coincidence degree theory or a fixed-point theorem on cones. We refer the reader to \cite{CM, FE, GNT,Ma1} for some recent results of nonlinear multipoint boundary-value problems. Let $a_i\geq 0$ for $i=1,\dots ,m-2$ and let $\xi_i$ satisfy $0<\xi _{1}<\xi_{2}<\dots <\xi_{m-2}<1$. Ma \cite{Ma2} applied a fixed-point theorem on cones to prove the existence of a positive solution of \begin{equation*} \begin{gathered} u''+a(t)f(u)=0\\ u(0)=0,\quad u(1)=\sum_{i=1}^{m-2}\alpha_iu(\xi_i) \end{gathered} \end{equation*} under superlinearity or sublinearity assumptions on $f$. He also assumed the following \begin{enumerate} \item[($\Gamma 1$)] $a\in C([0,1],[0,\infty ))$, $f\in C([0,\infty ),[0,\infty ))$, and there exists $t_{0}\in$ $[\xi_{m-2},1]$ such that $a(t_{0})>0$ \item[($\Gamma 2$)] For $i=1,\dots ,m-2$, $a_i\geq 0$ and $\sum_{i=1}^{m-2}a_i\xi_i<1$. \end{enumerate} Recently, Gupta \cite{GU1} obtained existence results for the boundary-value problem \begin{gather*} y''(t)=f(t,y(t),y'(t))+e(t),\quad 0\leq t\leq 1 \\ y(0)=0,\quad y(1)=\sum_{i=1}^{m-2}\alpha_iy(\xi_i), \end{gather*} by using the Leray-Schauder continuation theorem, under smallness assumptions of the form \begin{equation*} |f(t,y,y')| \leq p(t)|y| +q(t)| y'| +r(t)\quad\text{and} \quad C_{1}\| p(t)\| +C_{2}\|q(t)\| \leq1, \end{equation*} with $p(t)$, $q(t)$, $r(t)$ and $e(t)$ in $L^{1}(0,1)$ and $C_{1}$ and $C_{2}$ constants. In this paper, we consider the problem of existence of positive solutions for the m-point boundary-value problem \begin{gather} y''(t)=-f(t,y(t),y'(t)),\quad 0\leq t\leq1, \label{E} \\ \alpha y(0)-\beta y'(0)=0,\quad y(1)=\sum_{i=1}^{m-2}\alpha_iy(\xi_i). \label{C} \end{gather} We assume $\alpha>0$, $\beta>0$, the function $f$ is continuous, and $$f(t,y,y')\geq0,\quad \text{for all }t\in[0,1],\;y\geq 0\,\; y'\in\mathbb{R}. \label{A1}$$ The presence of the third variable $y'$ in the function $f(t,y,y')$ causes some considerable difficulties, especially, in the case where an approach relies on a fixed point theorem on cones and the growth rate of $f(t,y,y')$ is sublinear or superlinear. We overcome this predicament, by extending below the concept-assumptions (\ref{SL}) and ( \ref{sl}) as follows: Suppose that for any $M>0$, $$\begin{gathered} f_{0,0}:=\lim_{(y,y')\to (0,0)}\max_{0\leq t\leq1} \frac{f(t,y,y')}{y}=0 \\ f_{+\infty}:=\lim_{y\to +\infty}\min_{0\leq t\leq1}\frac {f(t,y,y')}{y}=+\infty, \quad\mbox{for }|y'|\leq M \end{gathered} \label{A2S}$$ i.e. $f$ is \emph{jointly superlinear} at the end point $(0,0)$ and \emph{ uniformly superlinear }at\emph{\ }$+\infty$. Similarly $$\begin{gathered} f_{0}:=\lim_{y\to 0+}\min_{0\leq t\leq1}\frac{f(t,y,y')} {y}=+\infty,\quad\mbox{for } |y'|\leq M.\\ f_{+\infty,+\infty}:=\lim_{(y,y')\to (+\infty,+\infty)} \max_{0\leq t\leq1}\frac{f(t,y,y')}{y}=0, \end{gathered} \label{A2s}$$ i.e. $f$ is \emph{jointly sublinear} at $(+\infty,+\infty)$ and \emph{ uniformly sublinear }at $0$. Furthermore there exist $\bar{l}\in(0,\infty]$, such that for every $\bar{M} >0$ $$\lim_{y'\to -\infty}\max_{0\leq t\leq1}\frac{f(t,y,y')}{ y'}=-\bar{l},\quad\text{for }y\in[0,\bar{M}] \label{A3}$$ i.e. $f(t,y,.)$ is \emph{linear or superlinear }at $-\infty$ \ and for every $\bar{\eta}>0$ $$\lim_{y'\to 0}\min_{0\leq t\leq1}\frac{f(t,y,y')}{y'} =0, \quad\text{for }y\in(0,\bar{\eta}). \label{A4}$$ i.e. $f(t,y,.)$ is \emph{superlinear }at $0$. \begin{remark}\label{R1} \rm Note that the differential equation (\ref{E}) defines a vector field whose properties will be crucial for our study. More specifically, we look at the $(y,y')$ face semi-plane $(y>0)$. From the sign condition on $f$ (see assumption (\ref{A1})), we immediately see that $y''<0$. Thus any trajectory $(y(t),y'(t))$, $t\geq0$, emanating from the semi-line $E_{0}:=\{(y,y'):\alpha y-\beta y'=0,\;y>0\}$ trends'' in a natural way, (when $y'(t)>0$) toward the positive $y$-semi-axis and then (when $y'(t)<0$) trends toward the negative $y'$-semi-axis. Lastly, by setting a certain growth rate on $f$ (say superlinearity) we can control the vector field, so that some trajectory satisfies the given boundary condition $y(1)=\sum_{i=1}^{m-2}\alpha_iy(\xi_i)$ at the time $t=1$. These properties will be referred as \emph{The nature of the vector field''} throughout the rest of paper. \end{remark} So the technique presented here is different to that given in the above mentioned papers \cite{GU1, EW,DEH, GNT,ET}, but it is closely related with those in \cite{JP,Ma2}. Actually, we rely on the above ''nature of the vector field'' and on the simple shooting method. Finally, for completeness we refer to the well-known Kneser's theorem (see for example Copel's text-book \cite{Co}). \begin{theorem} \label{Th1} Consider the system $$\;x''=f(t,x,x'),\quad (t,x,x')\in\Omega :=[\alpha,\beta]\times\mathbb{R}^{2n},\label{*}$$ with the function $f$ continuous. Let $\hat{E}_{0}$ be a continuum (compact and connected) set in $\Omega_{0}:=\{(t,x,x')\in \Omega:t=\alpha\}$ and let $\mathcal{X}(\hat{E}_{0})$ be the family of \ all solutions of (\ref{*}) emanating from $\hat{E}_{0}$. If any solution $x\in\mathcal{X}(\hat{E}_{0})$ is defined on the interval $[\alpha,\tau]$, then the set (\emph{cross-section }at the point $\tau$) $\mathcal{X}(\tau;\hat{E}_{0}):=\left\{ (x(\tau),x'(\tau )):x\in\mathcal{X}(\hat{E}_{0})\right\}$ is a continuum in $\mathbb{R}^{2n}$. \end{theorem} Now consider (\ref{E})-(\ref{C}) with the following notation. \begin{gather*} \sigma:=\sum_{i=1}^{m-2}\alpha_i\xi_i<1, \quad \sigma^{\ast}:=\sum_{i=1}^{m-2}\alpha_i\xi_i+\frac{\beta}{\alpha} \Big\{ \sum_{i=1}^{m-2}\alpha_i\frac{\xi_i}{\xi_{1}}-1\Big\} <1, \\ K_{0}:=\max\Big\{ \frac{2\alpha}{\beta},\;2\big[ \frac{\alpha+\beta}{\beta}- \frac{\sigma}{\xi_{m-2}}\big] \Big\} , \\ \mu_{0}:=\min\Big\{ (1-m^{\ast})\frac{\varepsilon\alpha}{\beta},\;2\big[ \frac{\varepsilon(\alpha+\beta)}{\beta}-1\big] \text{ }\Big\} \end{gather*} where $\beta/(\alpha+\beta)<\varepsilon<1$ and $\sigma^{\ast}0$ such that $$\min_{0\leq t\leq1}f(t,y,y')>-\bar{K}y',\;\;0\leq y\leq H \big( 1+\frac{\alpha}{\beta}\big)\quad\text{and}\quad y'<-H. \label{99}$$ By the superlinearity of $f(t,y,y')$ at $y=+\infty\;$(see condition ( \ref{A2S})), for any $K^{\ast}>K_{0}$ there exists $H^{\ast}>H$ such that $$\min_{0\leq t\leq1}f(t,y,y')>K^{\ast}y,\ y\geq H^{\ast}\quad\text{and } \quad -2H\leq y'\leq\frac{\alpha}{\beta}H. \label{100}$$ Similarly by the superlinearity of $f(t,y,y')$ at $(0,0)$, for any $0<\mu^{\ast}<\mu_{0}$ there is an $\eta^{\ast}>0$ such that 01.\label{MP*} \end{itemize} Furthermore, there exists a positive number $H$ such that $0K>K_{0} there exists H^{\ast }\geq H>0 such that $$\min_{0\leq t\leq 1}f(t,y,y')>Ky,\;\;\ y\geq H\quad \text{and}\quad \frac{\alpha }{\beta }H\geq y'\geq -2H. \label{1}$$ Consider the function \begin{equation*} W(P):=\sum_{i=1}^{m-2}\alpha_iy(\xi_i)-y(1), \end{equation*} where y\in \mathcal{X}(P_{1}) is any solution of differential equation ( \ref{E*}) starting at the point P_{1}:=(y_{1},y_{1}')\in E_{0} with y_{1}=H. By the assumption (\ref{A1}) (i.e. the nature of the vector field, see Remark \ref{R1}) it is obvious that y(t)\geq y_{1}=H and  y'(t)\leq y_{1}'=\frac{\alpha}{\beta}y_{1}=\frac{\alpha}{ \beta}H,\ for all t in a sufficiently small neighborhood of t=0. Let's suppose that there is t^{\ast}\in(0,1] such that \begin{equation*} y(t)\geq H,\;-2H\leq y'(t)\leq\frac{\alpha}{\beta}H,\;0\leq t\frac{ 2\alpha}{\beta}. Furthermore, by (\ref{2}), H\leq y(t)-\bar{K}y',\quad 0\leq y\leq H \big[ 1+\frac{\alpha}{\beta}\big] \quad\text{and}\quad y'<-H. \label{210} We shall prove that $$\frac{\alpha}{\beta}H\geq y'(t)\geq-\varepsilon^{\ast}H>-2H,\quad 0\leq t\leq1. \label{22}$$ Indeed, since y'(t) is decreasing on \left[ 0,1\right], let's assume that there exist t_{0},t_{1}\in(0,1) such that \begin{equation*} y'(t_{0})=-H\;,\;-\varepsilon^{\ast}H2\big[1+\frac{\alpha}{\beta}-\frac{\sigma}{\xi_{m-2}}\big]. \end{equation*} Similarly by the superlinearity of f(t,y,y') at (0,0), for any \mu>0 there is an \eta>0 such that 0\sum_{i=1}^{m-2}\alpha_i\xi_i+\frac{ \beta }{\alpha }\big\{\sum_{i=1}^{m-2}\alpha_i\frac{\xi_i}{\xi_{1}}-1 \big\}. It is now clear that the function W=W(P), P\in[P_{0},P_{1}] is continuous and thus by the Kneser's property (see Theorem \ref{Th1}), (\ref {5}) and (\ref{10}), we get a point P\in[P_{0},P_{1}] (we chose the last one to the left'' of P_{1}) such that W(P)=0. This fact clearly means that there is a solution y\in\mathcal{X}(P) of equation (\ref{E*}), such that \begin{equation*} W(P)=\sum_{i=1}^{m-2}\alpha_iy(\xi_i)-y(1)=0. \end{equation*} It remains to be proved that the so obtaining solution y=y(t) is actually a bounded function. Indeed, by the choice of P, the continuity of y(t) with respect initial values, (\ref{5}) and (\ref{10}), it follows that \begin{equation*} y(t)>0,\quad 0\leq t\leq1, \end{equation*} because if \begin{equation*} y(t)>0,\;0\leq t<1\quad\text{and}\quad y(1)=0, \end{equation*} then W(P)>0. Moreover by the nature of the vector field (see Remark \ref {R1}), there is t_{P}\in\left( 0,1\right)  such that the so obtaining solution y\in\mathcal{X}(P) is strictly increasing on \left[ 0,t_{p} \right] , strictly decreasing on \left[ t_{p},1\right]  and further is strictly positive on \left[ 0,1\right] . Also it holds y(t)\leq H, 0\leq t\leq1, i.e. 0 \frac{\alpha +\beta }{\beta } and recall that \begin{equation*} \sum_{i=1}^{m-2}\alpha_iy(\xi_i)+\frac{\beta }{\alpha }\Big\{ \sum_{i=1}^{m-2}\alpha_i\frac{\xi_i}{\xi_{1}}-1\Big\}0 such that $$\max_{0\leq t\leq 1}f(t,y,y')<\mu y,\quad y\geq H,\quad \text{and} \quad \frac{\alpha }{\beta }H\geq y'\geq m^{\ast }\frac{\alpha }{ \beta }H. \label{71}$$ Let's consider a point P_{0}:=(y_{0},y_{0}')\in E_{0} with  y_{0}=H. We will prove first that for any solution y\in \mathcal{X}(P_{0}) , $$H\leq y(t)\leq \varepsilon_{0}H\quad \text{and}\quad \frac{m^{\ast }\alpha }{\beta }H\leq y'(t)\leq \frac{\alpha }{\beta }H,\quad 0\leq t\leq 1. \label{72}$$ Let us suppose that this is not the case. Then by the assumption (\ref{A1}), there is t^{\ast }\in [ 0,1] such that \begin{gathered} H\leq y(t)\leq\varepsilon_{0}H,\quad \frac{m^{\ast}\alpha}{\beta }H\leq y'(t)\leq\frac{\alpha}{\beta}H,\quad 0\sum_{i=1}^{m-2}\alpha_i\xi_i+\frac{\beta }{\alpha}\big\{ \sum_{i=1}^{m-2}\alpha_i\frac{\xi_i}{\xi_{1}}-1\big\}. On the other hand, since f_{0}=+\infty , for any K>\max \{\frac{2(\alpha -\beta )}{\beta },\frac{2\alpha }{\beta }\} there exist \eta \in (0,H) such that \min_{0\leq t\leq 1}f(t,y,y')>Ky,\;0\max \{\frac{2(\alpha -\beta )}{\beta },\frac{ 2\alpha }{\beta }\} we can easily prove that $$\frac{\eta }{2}\leq y(t)\leq \eta ,\quad 0\leq t\leq 1. \label{76}$$ We choose now \varepsilon_{0}^{\ast }\in \left( 1,2\right)  and then\ by Assumption (\ref{A4}), there exist \bar{\eta}_{0}\in (0,\eta ) and 0-\eta ,\quad 0\leq t\leq 1. \label{78} Indeed since y'(t) is decreasing on \left[ 0,1\right]  and  \varepsilon_{0}^{\ast }\in \left( 1,2\right)  is arbitrary, let's assume that there exist t_{0},t_{1}\in [ 0,1] such that y^{\prime }(t_{0})=-\bar{\eta}_{0}, \begin{equation*} -2\bar{\eta}_{0}<-\varepsilon_{0}^{\ast }\bar{\eta}_{0}\leq y^{\prime }(t)\leq -\bar{\eta}_{0},\quad t_{0}\leq t\frac{y(\xi^{\ast})}{\xi^{\ast}},$ where clearly $\xi^{\ast}=\xi_{m-2}$ and this contradicts the concavity of the solution $y=y(t)$. Furthermore we must seek the monotone (obviously increasing) solutions of (\ref{E})-(\ref{C}), only for the case $\sum_{i=1}^{m-2}\alpha_i\geq1$, since otherwise we get $0=W(P)=\sum_{i=1}^{m-2}\alpha_iy(\xi_i)-y(1)<\Big[ \sum_{i=1} ^{m-2}\alpha_i-1\Big] y(1)<0.$ The question of existence of such a monotone solution remains open. However we can obtain a strictly decreasing solution for the boundary-value problem $$\begin{gathered} y''(t)=-f(t,y(t),y'(t)),\quad 0\leq t\leq1, \\ \alpha y(0)+\beta y'(0)=0,\quad y(1)=\sum_{i=1}^{m-2}\alpha_iy(\xi_i). \end{gathered} \label{C*}$$ where $\alpha\geq0$ $\$and $\beta>0$. \end{remark} \begin{remark} \label{R3} \rm Suppose that the concept of jointly sublinearity is modified to $$\begin{gathered} f_{0}:=\lim_{y\to 0+}\min_{0\leq t\leq1}\frac{f(t,y,y')} {y}=+\infty,\quad \mbox{for } |y'|\leq M.\\ f_{\infty,-\infty}:=\lim_{(y,y')\to (+\infty,-\infty)} \max_{0\leq t\leq1}\frac{f(t,y,y')}{y}=0. \end{gathered} \label{A2*}$$ Then, following almost the same line as above (under the obvious modifications) we may prove the next theorem. \end{remark} \begin{theorem} Assume that (\ref{A1}) holds and further $\sigma^{\ast}=\sum_{i=1}^{m-2}\alpha_i\xi_i+\frac{\beta}{\alpha} \Big\{\sum_{i=1}^{m-2}\alpha_i\frac{\xi_i}{\xi_{1}}-1\Big\} <1.$ Then the boundary-value problem (\ref{C*}) has a positive strictly decreasing solution provided that: \begin{itemize} \item The function $f$ is superlinear (see (\ref{A2S})) along with (\ref{A3}), or \item The function $f$ is sublinear (see (\ref{A2*})), (\ref{A4}) is true and in addition, $\sum_{i=1}^{m-2}\alpha_i\xi_i\big[ \frac{1}{\xi_{m-2}}-\frac{\alpha }{\beta}\big] >1.$ \end{itemize} Furthermore there exists a positive number $H$ such that $00.$ \end{remark} Finally consider the boundary-value problem $$\begin{gathered} y''+f(t,y,y')=0, y'(0)=0,\quad y(1)=\sum_{i=1}^{m-2}\alpha_iy(\xi_i). \end{gathered} \label{D}$$ Then following almost the same lines as above, we may prove the next theorem. \begin{theorem} Assume that (\ref{A1}) holds and $\sum_{i=1}^{m-2}\alpha_i\frac{\xi_i}{\xi_{1}}<1.$ Then the boundary-value problem (\ref{D}) has a positive strictly decreasing solution provided that \begin{itemize} \item The function $f$ is superlinear (see (\ref{A2S})) along with (\ref{A3}), or \item The function $f$ is sublinear (see (\ref{A2*})), (\ref{A4}) holds and in addition $\sum_{i=1}^{m-2}\alpha_i\frac{\xi_i}{\xi_{m-2}}>1.$ \end{itemize} Furthermore there exists a positive number $H$ such that \[ 0