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\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}
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\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}