\documentstyle[twoside]{article} \input amssym.def % used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Positive Solutions \hfil EJDE--1997/03}% {EJDE--1997/03\hfil Paul W. Eloe \& Johnny Henderson \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.\ {\bf 1997}(1997), No.\ 03, pp. 1--11. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113} \vspace{\bigskipamount} \\ Positive solutions and nonlinear multipoint conjugate eigenvalue problems \thanks{ {\em 1991 Mathematics Subject Classifications:} 34B10, 34B15.\newline\indent {\em Key words and phrases:} multipoint, nonlinear eigenvalue problem, cone. \newline\indent \copyright 1997 Southwest Texas State University and University of North Texas.\newline\indent Submitted December 17, 1996. Published January 22, 1997.} } \date{} \author{Paul W. Eloe \& Johnny Henderson} \maketitle \begin{abstract} Values of $\lambda$ are determined for which there exist solutions in a cone of the $n^{th}$ order nonlinear differential equation, $u^{(n)} = \lambda a(t) f(u)$, $0 < t < 1$, satisfying the multipoint boundary conditions, $u^{(j)}(a_i) = 0$, $0 \leq j \leq n_i -1$, $1 \leq i \leq k$, where $0 = a_1 < a_2 < \cdots < a_k = 1$, and $\sum _{i=1}^k n_i = n$, where $a$ and $f$ are nonnegative valued, and where both $\lim\limits_{|x| \to 0^+} f(x)/|x|$ and $\lim\limits_{|x| \to\infty} f(x)/|x|$ exist. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}{Remark}[section] \def\theequation{\thesection.\arabic{equation}} \newcommand{\I}{\int} \newcommand{\F}{\infty} \newcommand{\la}{\lambda} \newcommand{\refp}[1]{(\ref{#1})} \section{Introduction} Let $n \geq 2$ and $2 \leq k \leq n$ be integers, and let $0 = a_1 < a_2 < \cdots < a_k = 1$ be fixed. Also, let $n_1, \dots, n_k$ be positive integers such that $\sum \limits _{i=1}^k n_i = n$. We are concerned with determining eigenvalues, $\la$, for which there exist solutions, that are positive with respect to a cone, of the nonlinear multipoint conjugate boundary value problem, $$\label {e11} u^{(n)} = \la a(t)f(u), \quad 0 < t < 1,$$ $$\label {e12} u^{(j)} (a_i) = 0, \quad 0 \leq j \leq n_i -1, \quad 1 \leq i \leq k,$$ where \begin{itemize} \item [(A)] $f: \Bbb {R} \to [0, \F)$ is continuous, \item [(B)] $a:[0, 1] \to [0, \F)$ is continuous and does not vanish identically on any subinterval, and \item [(C)] $f_0 = \lim \limits _{|x| \to 0^+} \frac {f(x)}{|x|}$ and $f_{\F} = \lim \limits_{|x| \to \F} \frac {f(x)}{|x|}$ exist. \end{itemize} This work constitutes a complete generalization, in the conjugate problem setting, of the paper by Henderson and Wang \cite {HJ} which was devoted to the eigenvalue problem \refp {e11}, \refp {e12} for the case $n = 2$ and $k = 2$. While the paper \cite {HJ} arose from a cornerstone paper by Erbe and Wang \cite {EL}, which was devoted to $n = 2$ and $k = 2$ for the cases when $f$ is superlinear (i.e., $f_0 = 0$ and $f_{\F} = \F$) and when $f$ is sublinear (i.e., $f_0 = \F$ and $f_{\F} = 0$), the development since has been rapid. For example, Eloe and Henderson \cite {EP} gave a most general extension of \cite {HJ} for \refp {e11}, \refp {e12} in the case of arbitrary $n$ and $k = 2$. Other partial extensions have been given for higher order boundary value problems, as well as results for multiple solutions, in both the continuous and discrete settings; see for example \cite {RA,AR,PE,WP,EE,LE,JH,EK,WL,FM,MF}. Foundational work for this paper is the recent study by Eloe and McKelvey \cite {EO} of \refp {e11}, \refp {e12}, for arbitrary $n$, $k = 3$ and $n_1 = n_3 = 1$. For the case of $n = 2$ and $k = 2$, \refp {e11}, \refp {e12} describes many phenomena in the applied mathematical sciences such as, to name a few, nonlinear diffusion generated by nonlinear sources, thermal ignition of gases, and chemical concentrations in biological problems where only positive solutions are meaningful; see, for example \cite {DG,AF,KH,LS}. Higher order boundary value problems for ordinary differential equations arise naturally in technical applications. Frequently, these occur in the form of a multipoint boundary value problem for an $n^{th}$ order ordinary differential equation, such as an $n$-point boundary value problem model of a dynamical system with $n$ degrees of freedom in which $n$ states are observed at $n$ times; see Meyer \cite {GM}. It is noted in \cite {GM} that, strictly speaking, boundary value problems for higher order ordinary differential equations are a particular class of interface problems. One example in which this is exhibited is given by Keener \cite {JK} in determining the speed of a flagellate protozoan in a viscous fluid. Another particular case of a boundary value problem for a higher order ordinary differential equation arising as an interface problem is given by Wayner, {\em et al.} \cite {PC} in dealing with a study of perfectly wetting liquids. We now observe that, for $n = 2$, positive solutions of \refp {e11}, \refp {e12} are concave. This concavity was exploited in \cite {EL,HJ} and in many of the extensions cited above in defining a cone on which a positive operator was defined. A fixed point theorem due to Krasnosel'skii \cite {MK} was then applied to yield positive solutions for certain intervals of eigenvalues. In defining an appropriate cone, inequalities that provide lower bounds for positive functions as a function of the supremum norm have been applied. The inequality to which we refer may be stated as follows: {\em If $y \in C^{(2)}[0, 1]$ is such that $y(t) \geq 0$, $0 \leq t \leq 1$, and $y''(t) \leq 0$, $0 \leq t \leq 1$, then} $$\label {e13} y(t) \geq \frac {1}{4} \max _{0 \leq s \leq 1} |y(s)|, \quad \frac {1}{4} \leq t \leq \frac {3}{4}.$$ Inequality \refp {e13} was recently generalized by Eloe and Henderson \cite {ew} in the following sense: Let $n \geq 2$ and $2 \leq \ell \leq n-1$. If $y \in C^{(n)}[0, 1]$ is such that \begin{eqnarray*} (-1)^{n- \ell} y^{(n)}(t) \geq 0, \quad 0 \leq t \leq 1,\\ y^{(j)}(0) = 0, \quad 0 \leq j \leq \ell -1,\\ y^{(j)}(1) = 0, \quad 0 \leq j \leq n - \ell - 1, \end{eqnarray*} then $$\label {e14} y(t) \geq \frac {1}{4^m} \|y\|, \quad \frac {1}{4} \leq t \leq \frac {3}{4},$$ where $\|y\| = \max \limits_{0 \leq s \leq 1} |y(s)|$ and $m = \max \{\ell, n - \ell\}$.\\ An inequality analogous to \refp {e14} for a Green's function was also given in \cite {EP}. In a later paper, Eloe and Henderson \cite {PW} obtained a further generalization of \refp {e14} for solutions of differential inequalities satisfying the multipoint conjugate boundary conditions \refp {e12}. In that same paper \cite {PW}, an analogous inequality was also derived for a Green's function associated with $y^{(n)} = 0$ and \refp {e12}. It is that generalization of \refp {e14} as it applies to solutions of \refp {e11}, \refp {e12} which eventually leads to the main results of this paper. In Section 2, we state the generalization of \refp {e14} as it applies to solutions of \refp {e11}, \refp {e12}. We also state the analogous inequality for a Green's function that will be used in defining a positive operator on a cone. The Krasnosel'skii fixed point theorem is also stated in that section. Then, in Section 3, we give an appropriate Banach space and construct a cone on which we apply the fixed point theorem to our positive operator, thus yielding solutions of \refp {e11}, \refp {e12}, for open intervals of eigenvalues. \section{Preliminaries} \setcounter{equation}{0} In this section, we state the Krasnosel'skii fixed pointed theorem to which we referred in the introduction. Prior to this, we will state the generalization of \refp {e14} as given in \cite {PW}. For notational purposes, set $\alpha _i = \sum _{j=i + 1}^k n_j$, $1 \leq i \leq k-1$, let $S_i \subset (a_i, a_{i+1})$, $1 \leq i \leq k-1$, be defined by $$S_i = [(3a_i + a_{i+1})/4, (a_i + 3a_{i+1})/4],$$ let $$a = \min _{1 \leq i \leq k-1} \{a_{i+1} - a_i\},$$ and let $$m = \max \{n-n_1, n- n_k\}.$$ \begin{theorem} \label {t21} Assume $y \in C^{(n)}[0, 1]$ is such that $y^{(n)}(t) \geq 0$, $0 \leq t \leq 1$, and $y$ satisfies the multipoint boundary conditions \refp {e12}. Then, for each $1 \leq i \leq k-1$, $$\label {e21} (-1)^{\alpha _i} y(t) \geq \|y\| (\frac {a}{4})^m, \quad t \in S_i,$$ where $\|y\| = \max \limits_{0 \leq t \leq 1} |y(t)|$. \end{theorem} The Krasnosel'skii fixed point theorem will be applied to a completely continuous integral operator whose kernel, $G(t, s)$, is the Green's function for $$\label {e22} y^{(n)} = 0, \quad 0 \leq t \leq 1,$$ satisfying \refp {e12}. It is well-known \cite {WC} that $$\label {e23} (-1)^{\alpha _i} G(t, s) > 0 \mbox { on } (a_i, a_{i+1}) \times (0, 1), \quad 1 \leq i \leq k-1.$$ For the remainder of the paper, for $0 < s < 1$, let $\tau (s) \in (0, 1)$ be defined by $$\label {e24} |G(\tau (s), s)| = \sup _{0 \leq t \leq 1} |G(t, s)|,$$ so that, for each $1 \leq i \leq k-1$, $$\label {e25} (-1)^{\alpha _i} G(t, s) \leq |G(\tau (s), s)| \mbox { on } [a_i, a_{i+1}] \times [0, 1].$$ Then in analogy to \refp {e21}, Eloe and Henderson \cite {PW} proved the following inequality for $G(t, s)$. \begin{theorem} \label {t22} Let $G(t, s)$ denote the Green's function for \refp {e22}, \refp {e12}. Then, for $0 < s < 1$ and $1 \leq i \leq k-1$, $$\label {e26} (-1)^{\alpha _i} G(t, s) \geq (\frac {a}{4})^m |G(\tau (s), s)|, \quad t \in S_i.$$ \end{theorem} We mention that inequality \refp {e26} is closely related to inequalities derived for $G(t, s)$ by Pokornyi \cite {YP,PY}. Inequalities \refp {e25} and \refp {e26} are of fundamental importance in defining positive operators to which we will apply the following fixed point theorem \cite {MK}. \begin{theorem} \label {t23} Let ${\cal B}$ be a Banach space, and let ${\cal P} \subset {\cal B}$ be a cone in ${\cal B}$. Assume $\Omega _1$, $\Omega _2$ are open subsets of ${\cal B}$ with $0 \in \Omega _1 \subset \bar {\Omega}_1 \subset \Omega _2$, and let $$T: {\cal P} \cap (\bar {\Omega} _2 \backslash \Omega _1) \to {\cal P}$$ be a completely continuous operator such that, either \begin{itemize} \item [(i)] $\|Tu\| \leq \|u\|, u \in {\cal P} \cap \partial \Omega _1$, and $\|Tu\| \geq \|u\|, u \in {\cal P} \cap \partial \Omega _2$, or \item [(ii)] $\|Tu\| \geq \|u\|, u \in {\cal P} \cap \partial \Omega _1$, and $\|Tu\| \leq \|u\|, u \in {\cal P} \cap \partial \Omega _2$. \end{itemize} Then $T$ has a fixed point in ${\cal P} \cap (\bar {\Omega}_2 \backslash \Omega _1)$. \end{theorem} \section{Solutions in a Cone} \setcounter{equation}{0} In this section, we apply Theorem \ref {t23} to the eigenvalue problem \refp {e11}, \refp {e12}. The keys to satisfying the hypotheses of the theorem are in selecting a suitable cone and in inequalities \refp {e25} and \refp {e26}. As is standard, $u \in C[0, 1]$ is a solution of \refp {e11}, \refp {e12} if, and only if, $$u(t) = \lambda \I _0^1 G(t, s) a(s) f(u(s))ds, 0 \leq t \leq 1,$$ where $G(t, s)$ is the Green's function for \refp {e22}, \refp {e12}. We let ${\cal B} = C[0, 1]$, and for $y \in {\cal B}$, define $\|y\| = \sup _{0 \leq t \leq 1} |y(t)|$. Then $({\cal B}, \|\cdot\|)$ is a Banach space. The cone, ${\cal P}$, in which we shall exhibit solutions is defined by $$\begin{array}{c} {\cal P} = \{x \in {\cal B} \mid \mbox { for } 1 \leq i \leq k-1, (-1) ^{\alpha _i} x(t) \geq 0 \mbox { on } [a_i, a_{i+1}],\\ \mbox { and } \min _{t \in S_i} (-1)^{\alpha _i} x(t) \geq (\frac {a}{4}) ^m \|x\|\}. \end{array}$$ \begin{theorem} \label {t31} Assume that conditions (A), (B) and (C) are satisfied. Then, for each $\lambda$ satisfying, $$\label {e31} \frac {4^m}{a^m \sum _{i=1}^{k-1} \I_{S_i} |G(\frac {1}{2}, s)|a(s) dsf_{\F}} < \lambda < \frac {1}{\I_0^1 |G(\tau (s), s)|a(s)ds f_0},$$ there is at least one solution of \refp {e11}, \refp {e12} belonging to ${\cal P}$. \end{theorem} \paragraph{Proof} We remark that a special case in the arguments result when $f_{\F} = \F$. However, the modifications required for that case, in the following proof, are straightforward, and so we omit those details. Let $\lambda$ be given as in \refp {e31}, and let $\epsilon > 0$ be such that $$\frac {4^m}{a^m \sum _{i=1}^{k-1} \I_{S_i} |G(\frac {1}{2}, s)|a(s)ds(f_{\F} - \epsilon)} \leq \lambda \leq \frac {1}{\I_0^1 |G(\tau (s), s)|a(s)ds(f_0 + \epsilon)}.$$ We seek a fixed point of the integral operator $T: {\cal P} \to {\cal B}$ defined by $$\label {e32} Tu(t) = \lambda \I_0^1 G(t, s) a(s) f(u(s)) ds, \quad u \in {\cal P}.$$ First, let $u \in {\cal P}$ and let $t \in [0, 1]$. Then, for some $1 \leq i \leq k-1$, we have $t \in [a_i, a_{i+1}]$, and by \refp {e23} and \refp {e25}, \begin{eqnarray*} 0 \leq (-1)^{\alpha _i} Tu(t) & = & \lambda \I_0^1 (-1) ^{\alpha _i} G(t, s) a(s) f(u(s))ds\\ & \leq & \lambda \I_0^1 |G(\tau (s), s)|a(s)f(u(s))ds, \end{eqnarray*} so that $$\label {e33} \|Tu\| \leq \lambda \I_0^1 |G(\tau (s), s)|a(s)f(u(s))ds.$$ Moreover, for $u \in {\cal P}$ and $t \in S_i$, $1 \leq i \leq k-1$, we have from \refp {e26} and \refp {e33}, \begin{eqnarray*} \min _{t \in S_i} (-1)^{\alpha_i} Tu(t) & = & \min _{t \in S_i} \lambda \I_0^1 (-1) ^{\alpha _i} G(t, s) a(s) f(u(s))ds\\ & \geq & (\frac {a}{4})^m \lambda \I_0^1 |G(\tau (s), s)| a(s) f(u(s))ds\\ & \geq & (\frac {a}{4})^m \|Tu\|. \end{eqnarray*} As a consequence $T: {\cal P} \to {\cal P}$. The standard arguments can also be used to verify that $T$ is completely continuous. We begin with $f_0$. There exists an $H_1 > 0$ such that $f(x) \leq (f_0 + \epsilon) |x|$, for $0 < |x| < H_1$. So, if we choose $u \in {\cal P}$ with $\|u\| = H_1$, then from \refp {e25} \begin{eqnarray*} |Tu(t)| & \leq & \lambda \I_0^1 |G(\tau (s), s)|a(s) f(u(s))ds\\ & \leq & \lambda \I_0^1|G(\tau (s), s)|a(s)(f_0 + \epsilon)|u(s)|ds\\ & \leq & \lambda \I_0^1 |G(\tau (s), s)|a(s)ds(f_0 + \epsilon) \|u\| \\ & \leq & \|u\|, 0 \leq t \leq 1. \end{eqnarray*} So, $\|Tu\| \leq \|u\|$. We set $$\Omega _1 = \{x \in {\cal B} \mid \|x\| < H_1\}.$$ Then $$\label {e34} \|Tu\| \leq \|u\|, \mbox { for } u \in {\cal P} \cap \partial \Omega _1.$$ Next, we consider $f_{\F}$. There exists an $\bar {H}_2 > 0$ such that $f(x) \geq (f_{\F} - \epsilon) |x|$, for all $|x| \geq \bar {H}_2$. Let $H_2 = \max \{2H_1, (\frac {4}{a})^m \bar {H}_2\}$, and define $$\Omega _2 = \{x \in {\cal B} \mid \|x\| < H_2\}.$$ Let $u \in {\cal P}$ with $\|u\| = H_2$. Then, for each $1 \leq i \leq k-1$, $\min _{t \in S_i} (-1)^{\alpha _i} u(t) \geq (\frac {a}{4})^m \|u\| \geq \bar {H}_2$. Moreover, there exists $1 \leq i_0 \leq k-1$ such that $\frac {1}{2} \in [a_{i_0}, a_{i_0+1}]$. Then, by \refp {e23}, \begin{eqnarray*} (-1)^{\alpha _{i_0}} Tu(\frac {1}{2}) & = & \lambda \I_0^1 (-1)^{\alpha _{i_0}} G(\frac {1}{2}, s) a(s)f(u(s))ds\\ & = & \lambda \I_0^1 |G(\frac {1}{2}, s)| a(s) f(u(s))ds\\ & \geq & \lambda \sum _{i=1}^{k-1} \I_{S_i} |G(\frac {1}{2}, s)|a(s) f(u(s))ds\\ & \geq & \lambda \sum _{i=1}^{k-1} \I_{S_i} |G(\frac {1}{2}, s)| a(s) (f_{\F} - \epsilon)|u(s)|ds\\ & \geq & \lambda (\frac {a}{4})^m \sum _{i=1}^{k-1} \I_{S_i} |G(\frac {1}{2}, s)|a(s)ds (f_{\F} - \epsilon) \|u\| \\ & \geq & \|u\|. \end{eqnarray*} Thus, $\|Tu\| \geq \|u\|$. Hence, $$\label {e35} \|Tu\| \geq \|u\|, \mbox { for } u \in {\cal P} \cap \partial \Omega _2.$$ We apply part (i) of Theorem \ref {t23} in obtaining a fixed point, $u$, of $T$ that belongs to ${\cal P} \cap (\bar {\Omega}_2 \backslash \Omega _1)$. The fixed point, $u$, is a desired solution of \refp {e11}, \refp {e12}, for the given $\lambda$. The proof is complete. \hfil$\Box$ \begin{remark} \label {r1} It follows from Theorem \ref {t31}, if $f$ is superlinear (i.e., $f_0 = 0$ and $f_{\F} = \F$), then \refp {e11}, \refp {e12} has a solution, for each $0 < \la < \F$. \end{remark} \begin{theorem} \label {t32} Assume that conditions (A), (B) and (C) are satisfied . Then, for each $\la$ satisfying $$\label {e36} \frac {4^m}{a^m \sum _{i=1}^{k-1} \I_{S_i} |G(\frac {1}{2}, s)|a(s) ds f_0} < \la < \frac {1}{\I_0^1 |G(\tau (s), s)|a(s)ds f_{\F}},$$ there is at least one solution of \refp {e11}, \refp {e12} belonging to ${\cal P}$. \end{theorem} \paragraph{Proof} Let $\la$ be as in \refp {e36}, and choose $\epsilon > 0$ such that $$\frac {4^m}{a^m \sum _{i=1}^{k-1} \I_{S_i} |G(\frac {1}{2}, s)|a(s)ds (f_0 - \epsilon)} \leq \la \leq \frac {1}{\I_0^1 |G(\tau (s), s)|a(s) ds(f_{\F} + \epsilon)}.$$ Let $T$ be the cone preserving, completely continuous operator that was defined by \refp {e32}. Beginning with $f_0$, there exists an $H_1 > 0$ such that $f(x) \geq (f_0 - \epsilon) |x|$, for $0 < |x| \leq H_1$. Choose $u \in {\cal P}$ with $\|u\| = H_1$. As in Theorem \ref {t31}, there exists $1 \leq i_0 \leq k-1$ such that $\frac {1}{2} \in [a_{i_0}, a_{i_0 +1}]$. Then \begin{eqnarray*} (-1)^{\alpha _{i_0}} Tu(\frac {1}{2}) & = & \la \I_0^1 (-1) ^{\alpha _{i_0}} G(\frac {1}{2}, s)a(s) f(u(s))ds\\ & = & \la \I_0^1 |G(\frac {1}{2}, s)| a(s) f(u(s))ds\\ & \geq & \la \sum _{i=1}^{k-1} \I_{S_i} |G(\frac {1}{2}, s)|a(s) f(u(s))ds\\ & \geq & \la \sum _{k=1}^{k-1} \I_{S_i} |G(\frac {1}{2}, s)|a(s) (f_0 - \epsilon) |u(s)|ds\\ & \geq & \la (\frac {a}{4})^m \sum _{i=1}^{k-1} \I_{S_i} |G(\frac {1}{2}, s)|a (s)ds (f_0 - \epsilon) \|u\| \\ & \geq & \|u\|. \end{eqnarray*} Therefore, if we let $$\Omega _1 = \{x \in {\cal B} \mid \|x\| < H_1\},$$ then $$\label {e37} \|Tu\| \geq \|u\|, \mbox { for } u \in {\cal P} \cap \partial \Omega _1.$$ We now consider $f_{\F}$. There exists an $\bar {H}_2 > 0$ such that $f(x) \leq (f_{\F} + \epsilon)|x|$, for all $|x| \geq \bar {H}_2$. There are the two cases, (a) $f$ is bounded, or (b) $f$ is unbounded. For (a), suppose $N > 0$ is such that $f(x) \leq N$, for all $x \in \Bbb {R}$. Let $H_2 = \max \{2H_1$,\\ $N \la \I_0^1 |G(\tau (s), s)|a(s)ds\}$. Then, for $u \in {\cal P}$ with $\|u\| = H_2$, \begin{eqnarray*} |Tu(t)| & \leq & \la \I_0^1 |G(t, s)| a(s) f(u(s))ds\\ & \leq & \la N \I_0^1 |G(\tau (s), s)| a(s) ds\\ & \leq & \|u\|, \quad 0 \leq t \leq 1. \end{eqnarray*} Thus, $\|Tu\| \leq \|u\|$. So, if $$\Omega _2 = \{x \in {\cal B} \mid \|x\| < H_2\},$$ then $$\label {e38} \|Tu\| \leq \|u\|, \mbox { for } u \in {\cal P} \cap \partial \Omega _2.$$ For case (b), let $H_2 > \max \{2H_1, \bar {H}_2\}$ be such that $f(x) \leq f(H_2)$, for $0 < |x| \leq H_2$. Let $u \in {\cal P}$ with $\|u\| = H_2$, and choose $t \in [0, 1]$. Then, for some $1 \leq i \leq k-1$, $t \in [a_i, a_{i+1}]$, and by \refp {e25}, \begin{eqnarray*} (-1)^{\alpha _i} Tu(t) & = & \la \I_0^1 (-1) ^{\alpha _i} G(t, s) a(s)f(u(s))ds\\ & = & \la \I_0^1 |G(t, s)| a(s) f(u(s))ds\\ & \leq & \la \I _0^1 |G(\tau (s), s)|a(s) f(H_2)ds\\ & \leq & \la \I_0^1 |G(\tau (s), s)| a(s)ds (f_{\F} + \epsilon) H_2\\ & = & \la \I_0^1 |G(\tau (s), s)|a(s) ds( f_{\F} + \epsilon) \|u\| \\ & \leq & \|u\|, \end{eqnarray*} so that $\|Tu\| \leq \|u\|$. For this case, if we let $$\Omega _2 = \{x \in {\cal B} \mid \|x\| < H_2\},$$ then $$\|Tu\| \leq \|u\|, \mbox { for } u \in {\cal P} \cap \partial \Omega _2.$$ Thus, regardless of the cases, an application of part (ii) of Theorem \ref {t23} yields a fixed point of $T$ which belongs to ${\cal P} \cap (\bar {\Omega}_2 \backslash \Omega _1)$. This fixed point is a solution of \refp {e11}, \refp {e12} corresponding to the given $\la$. 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Eloe\\ Department of Mathematics\\ University of Dayton\\ Dayton, Ohio 45469-2316 USA}\\ E-mail address: eloe@saber.udayton.edu \medskip {\sc Johnny Henderson\\ Department of Mathematics\\ Auburn University\\ Auburn, AL 36849 USA}\\ E-mail address: hendej2@mail.auburn.edu \end{document}