\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,

\begin{equation}  \label {e11}
u^{(n)} = \la a(t)f(u), \quad  0 < t < 1,
\end{equation}
\begin{equation}  \label {e12}
u^{(j)} (a_i) = 0, \quad  0 \leq j \leq n_i -1, \quad  1 \leq i \leq k,
\end{equation}
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}
\begin{equation}  \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}.
\end{equation}
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 
\begin{equation}  \label {e14}
y(t) \geq \frac {1}{4^m} \|y\|, \quad  \frac {1}{4} \leq t \leq \frac {3}{4},
\end{equation}
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 $1 \leq i \leq k-1$,

\begin{equation}  \label {e21}
(-1)^{\alpha _i} y(t) \geq \|y\| (\frac {a}{4})^m, \quad t \in S_i,
\end{equation}
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 

\begin{equation}  \label {e22}
y^{(n)} = 0, \quad  0 \leq t \leq 1,
\end{equation}
satisfying \refp {e12}.  It is well-known \cite {WC} that 

\begin{equation}  \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.
\end{equation}
For the remainder of the paper, for $0 < s < 1$, let $\tau (s) \in (0,
1)$ be defined by

\begin{equation}  \label {e24}
|G(\tau (s), s)| = \sup _{0 \leq t \leq 1} |G(t, s)|,
\end{equation}
so that, for each $1 \leq i \leq k-1$,

\begin{equation}  \label {e25}
(-1)^{\alpha _i} G(t, s) \leq |G(\tau (s), s)| \mbox { on } [a_i,
a_{i+1}] \times [0, 1].
\end{equation}

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 $1 \leq i \leq k-1$,

\begin{equation}  \label {e26}
(-1)^{\alpha _i} G(t, s) \geq (\frac {a}{4})^m |G(\tau (s), s)|, \quad
t \in S_i.
\end{equation}
\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,

\begin{equation}  \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}, 
\end{equation}
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

\begin{equation}  \label {e32}
Tu(t) = \lambda \I_0^1 G(t, s) a(s) f(u(s)) ds, \quad u \in {\cal P}.
\end{equation}

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

\begin{equation}  \label {e33}
\|Tu\| \leq \lambda \I_0^1 |G(\tau (s), s)|a(s)f(u(s))ds.
\end{equation}
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

\begin{equation}  \label {e34}
\|Tu\| \leq \|u\|, \mbox { for } u \in {\cal P} \cap \partial \Omega
_1.
\end{equation}

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,

\begin{equation}  \label {e35}
\|Tu\| \geq \|u\|, \mbox { for } u \in {\cal P} \cap \partial \Omega
_2.
\end{equation}
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

\begin{equation}  \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}},
\end{equation}
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

\begin{equation}  \label {e37}
\|Tu\| \geq \|u\|, \mbox { for } u \in {\cal P} \cap \partial \Omega
    _1.
\end{equation}

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

\begin{equation}  \label {e38}
\|Tu\| \leq \|u\|, \mbox { for } u \in {\cal P} \cap \partial \Omega
_2.
\end{equation}

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
$\la$.  The proof is complete.
\hfil$\Box$

\begin{remark}  \label {r2}
We observe that, if $f$ is sublinear (i.e., $f_0 = \F$ and $f_{\F} =
0$), then Theorem \ref {t32} yields a solution of \refp {e11}, \refp
{e12}, for all $0 < \la < \F$.
\end{remark}

\begin{thebibliography}{99}
\bibitem {RA} R. P. Agarwal and P. J. Y. Wong, Eigenvalues of boundary
  value problems for higher order differential equations, {\em
  Math. Problems in Eng.}, in press.

\bibitem {AR} R. P. Agarwal and P. J. Y. Wong, Eigenvalue
  characterization for $(n, p)$ boundary value problems, {\em
  J. Austral. Math. Soc. Ser. B: Appl. Math.}, in press.

\bibitem {WC} W. Coppel, {\em Disconjugacy}, Lecture Notes in
  Mathematics, vol. 220, Springer-Verlag, Berlin and New York, 1971.

\bibitem {PE} P. W. Eloe and J. Henderson, Positive solutions for
  $(n-1, 1)$ conjugate boundary value problems, {\em Nonlin. Anal.},
  in press.

\bibitem {ew} P. W. Eloe and J. Henderson, Inequalities based on a
  generalization of concavity, {\em Proc. Amer. Math. Soc.}, in press.

\bibitem {EP} P. W. Eloe and J. Henderson, Posititive solutions and
  nonlinear $(k, n-k)$ conjugate eigenvalue problems, {\em
  Diff. Eqn. Dyn. Sys.} in press.

\bibitem {PW} P. W. Eloe and J. Henderson, Inequalities for solutions
  of multipoint boundary value problems, preprint.

\bibitem {WP} P. W. Eloe, J. Henderson and E. R. Kaufmann, Multiple
  positive solutions for difference equations, preprint.

\bibitem {EE} P. W. Eloe, J. Henderson and P. J. Y. Wong, Positive
  solutions for two-point boundary value problems, {\em
  Dyn. Sys. Appl.} {\bf 2} (1996), 135-144.

\bibitem {EO} P. W. Eloe and J. McKelvey, Positive solutions of three
  point boundary value problems, {\em Comm. Appl. Nonlin. Anal.} {\bf
  4} (1997), in press.

\bibitem {LE} L. H. Erbe, S. Hu and H. Wang, Multiple positive
  solutions of some boundary value problems, {\em J. Math. Anal. Appl.}
  {\bf 184} (1994), 640-648.

\bibitem {EL} L. H. Erbe and H. Wang, On the existence of positive
  solutions of ordinary differential equations, {\em Proc.
  Amer. Math. Soc.} {\bf 120} (1994), 743-748.

\bibitem {DG} D. G. de Figueiredo, P. L. Lions and R. O. Nussbaum,
  {\em A priori} estimates  and existence of positive solutions of
  semilinear elliptic equations, {\em J. Math. Pura Appl.} {\bf 61}
  (1982), 41-63.

\bibitem {AF} A. M. Fink and G. A. Gatica, Positive solutions of
  second order systems of boundary value problems, {\em
  J. Math. Anal. Appl.} {\bf 180} (1993), 93-108.

\bibitem {JH} J. Henderson and E. R. Kaufmann, Multiple positive
  solutions for focal boundary value problems, {\em Comm. Appl. Anal.}
  {\bf 1} (1997), 53-60.

\bibitem {HJ} J. Henderson and H. Wang, Positive solutions for a
  nonlinear eigenvalue problem, {\em J. Math. Anal. Appl.}, in press.

\bibitem {EK} E. R. Kaufmann, Multiple positive solutions for higher
  order boundary value problems, {\em Rocky Mtn. J. Math.}, in press.

\bibitem {JK} J. P. Keener, {\em Principles of Applied Mathematics},
  Addison-Wesley, Redwood City, CA, 1988.

\bibitem {MK} M. A. Krasnosel'skii, {\em Positive Solutions of
    Operator Equations}, Noordhoff, Groningen, 1964.

\bibitem {KH} H. J. Kuiper, On positive solutions of nonlinear
  elliptic eigenvalue problems, {\em Rend. Math. Circ. Palermo}, Serie
  II, Tom. {\bf XX} (1979), 113-138.

\bibitem {WL} W. C. Lian, F. H. Wong and C. C. Yeh, On the existence
  of positive solutions of nonlinear second order differential
  equations, {\em Proc. Amer. Math. Soc.} {\bf 124} (1996), 1117-1126.

\bibitem {FM} F. Merdivenci, Green's matrices and positive solutions
  of a discrete boundary value problem, {\em Panamer. Math. J.} {\bf
  5} (1995), 25-42.

\bibitem {MF} F. Merdivenci, Two positive solutions of a boundary
  value problem for difference equations, {\em J. Difference
  Eqns. Appl.} {\bf 1} (1995), 263-270.

\bibitem {GM} G. H. Meyer, {\em Initial Value Methods for Boundary
    Value Problems}, Academic Press, New York, 1973.

\bibitem {YP} Yu. V. Pokornyi, On estimates for the Green's function
  for a multipoint boundary problem, {\em Mat. Zametki} {\bf 4}
  (1968), 533-540 (English translation).

\bibitem {PY} Yu. V. Pokornyi, Second solutions of a nonlinear de la
  Vall\'{e}e Poussin problem, {\em Diff. Urav.} {\bf 6} (1970),
  1599-1605 (English translation).

\bibitem {LS} L. Sanchez, Positive solutions for a class of semilinear
  two-point boundary value problems, {\em Bull. Austral. Math. Soc.}
  {\bf 45} (1992), 439-451.

\bibitem {PC} P. C. Wayner, Y. K. Kao and L. V. LaCroix, The interline
  heat transfer coefficient of an evaporating wetting film, {\em
  Int. J. Heat Mass Transfer} {\bf 19} (1976), 487-492.
\end{thebibliography}

\bigskip

{\sc Paul W. 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}