\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small \emph{
Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 55, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2011/55\hfil Integral boundary conditions]
{Positive solutions of a boundary value problem with integral boundary
conditions}
\author[J. R. L. Webb\hfil EJDE-2011/55\hfilneg]
{Jeff R. L. Webb}
\address{Jeff R. L. Webb\newline
School of Mathematics and Statistics,
University of Glasgow, Glasgow G12 8QW, UK}
\email{Jeffrey.Webb@glasgow.ac.uk}
\thanks{Submitted February 25, 2011. Published May 2, 2011.}
\subjclass[2000]{34B10, 34B15, 34B18}
\keywords{Nonlocal boundary conditions; positive solution}
\begin{abstract}
We consider boundary-value problems studied in a recent paper.
We show that some existing theory developed by Webb and
Infante applies to this problem and we use the known
theory to show how to find improved estimates on parameters
$\mu^*, \lambda^*$ so that some nonlinear differential
equations, with nonlocal boundary conditions of integral type,
have two positive solutions for all $\lambda$ with
$\mu^*< \lambda < \lambda^*$.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\section{Introduction}
In a recent paper \cite{C-T-ejde2011}, Chasreechai and Tariboon
gave some existence theorems for positive solutions of problems
of the following type.
\begin{equation}
\label{eq1}
u''(t)+{\lambda}g(t)f(u(t))=0, \quad t \in(0,1),
\end{equation}
with nonlocal boundary conditions (BCs) of the form
\begin{equation}
\label{bc1}
u(0)=\beta_1\int_0^\eta u(s)\,ds, \quad
u(1)=\beta_2\int_0^\eta u(s)\,ds.
\end{equation}
Here $\eta \in (0,1)$ is given and $\beta_1,\beta_2$ are to
satisfy some inequalities. The authors of \cite{C-T-ejde2011}
actually studied the problem on an interval $[0,T]$
but, since, without loss of generality, most previous theory has
considered the interval $[0,1]$, we have changed notation to the
usual one.
The authors of \cite{C-T-ejde2011} were unaware that this
equation, and others, with more general BCs had already been
studied in
\cite{jw-gi-jlms-06,zh-yang-na06,jw-gi-jlms-09,jw-dsa-09,bouch-na-09}.
In fact, a general theory which applies to many problems was given
in \cite{jw-gi-jlms-06}. There the authors studied in detail
\eqref{eq1} with the nonlocal BCs involving linear functionals
$\beta_i[u]$, that is, Riemann-Stieltjes integrals
\begin{equation}
\label{bc2}
u(0)=\beta_1[u]:=\int_0^1 u(s)\,dB_1(s), \quad
u(1)=\beta_2[u]:=\int_0^1 u(s)\,dB_2(s),
\end{equation}
where $B_j$ can be functions of bounded variation, equivalently
$dB_1, dB_2$ are \emph{signed} measures. Clearly the BCs
in \eqref{bc1} are special cases of the BCs in \eqref{bc2}.
The paper \cite{jw-gi-jlms-09} extended this to allow equations
of higher order with BCs of a similar type. Using the theory
of \cite{jw-gi-jlms-06} reduces the problem of showing there
are positive solutions to that of calculating explicitly the
constants that occur in that theory.
Problems with BCs of Riemann-Stieltjes type have been studied by
Karakostas and Tsamatos, \cite{kt-tmna,kt-ejde}, and by Yang
\cite{zh-yang-na06, zh-yang-na08}, and with a different method,
mixed monotone operator theory, by Kong \cite{l-kong-na10}, all
with (positive) measures. The special case of multipoint BCs has
been very extensively studied, for example
\cite{gnt-jmaa,j-l-ejde09,mt-jmaa}; for a survey of much work up
to 2007 see \cite{r-ma-survey}. The case of BCs of integral type
is also well studied, for example, \cite{bno-bvp11,bouch-na-09}.
The idea of having Riemann-Stieltjes type BCs with
\emph{sign changing} measures is due to
Webb-Infante \cite{jw-gi-nodea08,jw-gi-jlms-06}.
A typical examples is when
\begin{equation*}
\beta[u]=\sum_{i=1}^{m} \beta_{i}u(\xi_i)+\int_0^1 b(s)u(s)\,ds,
\end{equation*}
with $\xi_{i} \in (0,1)$, under some suitable positivity
assumptions, but, in the general case, some coefficients $\beta_i$
can be negative and $b$ can take some negative values.
Let $f^{0}:=\lim_{u \to 0}f(u)/u$,
$f^{\infty}:=\lim_{u \to \infty}f(u)/u$,
(with $\infty$ an allowed `value').
The results of \cite{C-T-ejde2011} are of the following type
\par
\begin{itemize}
\item[(1)]
For every $\lambda>0$ the boundary value problem (BVP) \eqref{eq1}, \eqref{bc1} has at least one positive solution if
\begin{gather*}
\text{either } f^{0}=0 \text{ and } f^{\infty}=\infty, \quad
\text{(superlinear case)}, \\
\text{or } f^{0}=\infty \text{ and } f^{\infty}=0, \quad
\text{(sublinear case)}.
\end{gather*}
\item[(2)] If $f^{0}=\infty$ and $f^{\infty}=\infty$ and there
is $\rho_1$ such that $\lambda f(u) \le \rho_1/\Lambda_1$
for all $u \in [0, \rho_1]$,
($\Lambda_1$ is a constant determined by the problem),
then the boundary value problem (BVP) \eqref{eq1}, \eqref{bc1}
has at least two positive solutions.
\item[(3)]
If $f^{0}=0$ and $f^{\infty}=0$ and there is $\rho_2$ such
that $\lambda f(u) \ge \rho_1/\Lambda_2$ for all
$u \in [c\rho_2, \rho_2]$, ($c,\Lambda_2$ are constants
determined by the problem), then the boundary value
problem (BVP) \eqref{eq1}, \eqref{bc1} has at least two
positive solutions.
\end{itemize}
Results of the same type had been given in
\cite{jw-gi-jlms-06,jw-gi-jlms-09}.
In \cite{C-T-ejde2011} Krasnosel'ski\u\i{}'s theorem
(see, for example, \cite{kr-zab}) is used, whereas
\cite{jw-gi-jlms-06,jw-gi-jlms-09} uses fixed point index theory,
in particular, some results of \cite{jw-kql-tmna}.
For a given nonlinear term $f$ the conditions in $(2)$ are of the form:
\begin{quote}
There exists $\lambda^*$ such that at least two positive solutions exist
for every $\lambda$ with $\lambda \le \lambda^*$.
\end{quote}
The conditions in (3) are of the form:
\begin{quote}
There exists $\lambda^{**}$ such that at least two positive
solutions exist
for every $\lambda$ with $\lambda \ge \lambda^{**}$.
\end{quote}
It is natural to ask whether $\lambda^*, \lambda^{**}$
can be closely estimated.
We intend to give a detailed account of the
BVP \eqref{eq1}, \eqref{bc1} and we shall show how the
results of \cite{jw-gi-jlms-06}, and some nonexistence results
from \cite{jw-jlms-10}, can be used to determine some
explicit \emph{upper and lower} bounds for these quantities.
Indeed we show for $(2)$ that there is an explicit $\mu^*$
such that there is no positive solution for $\lambda > \mu^*$
and determine an explicit $\lambda^*$. Similarly we find
explicit upper and lower bounds in case $(3)$.
The BCs \eqref{bc1} are rather special, the second BC is a
constant multiple of the first, accordingly the calculations
using the methods of \cite{jw-gi-jlms-06,jw-gi-jlms-09} simplify
dramatically, in fact we give a shortcut that applies in this case.
We obtain some new estimates of constants related to the constant
$c$ in (3) above.
We illustrate our results by revisiting the examples
in \cite{C-T-ejde2011} and show that our estimates on $\lambda$
in these examples give upper and lower bounds that are quite
close together. Our estimates substantially improve those
of \cite{C-T-ejde2011}.
We also use a result of \cite{jw-jlms-10}, which applies in this case,
and gives a slightly stronger conclusion on one example.
\section{Summary of some known theory}
We work throughout in the Banach space $C[0,1]$ of continuous
real-valued functions defined on $[0,1]$ endowed with the
norm $\|u\|:=\max_{t \in [0,1]}|u(t)|$.
We recall the theory developed in \cite{jw-gi-jlms-06} as it
applies to the following nonlocal BVP,
\begin{equation}
\label{eq-bvp}
\begin{gathered}
u''(t)+g(t)f(u(t))=0, \quad t \in (0,1), \\
u(0)=\beta_1[u], \quad u(1)=\beta_2[u],
\end{gathered}
\end{equation}
where $\beta_j[u]$ are linear functionals on $C[0,1]$, hence given by
\begin{equation}
\label{eq-rsbc}
\beta_1[u]=\int_0^1 u(t)\,dB_1(t), \quad
\beta_2[u]=\int_0^1 u(t)\,dB_2(t),
\end{equation}
where $B_j$ are functions of bounded variation.
We suppose for simplicity, that $f$ is continuous, $g$ satisfies
an integrability condition $g \Phi \in L^1$, so $g$ can have
pointwise singularities, where $\Phi$ is given below; for more
general conditions see \cite{jw-gi-jlms-06,jw-gi-jlms-09}.
Here we will only consider the case when $\beta_j$ are positive
functionals.
The first step is to find the Green's function $G$ for the problem.
A general method for doing this is given in
\cite{jw-gi-jlms-06,jw-gi-jlms-09} and is recalled below.
Positive solutions of the BVP are then equivalent to positive
fixed points of the integral operator
\begin{equation}
\label{eq-nonintop}
Su(t)=\int_0^1 G(t,s) g(s)f(u(s))\,ds.
\end{equation}
To find positive solutions we seek fixed points of the operator $S$
in some sub-cone $K$ of the cone
\begin{equation*}
P:=\{u \in C[0,1]: u(t) \ge 0, \text{ for } t \in [0,1]\}.
\end{equation*}
The theory in \cite{jw-gi-jlms-06} requires the following condition.
\begin{itemize}
\item[(C)] There exist a subinterval $[a,b] \subseteq [0,1]$,
a measurable function
$\Phi$, and a constant $c(a,b) >0$ such that
\begin{gather*}
G(t,s)\leq \Phi(s) \text{ for } t \in [0,1] \text{ and } s\in [0,1]\\
G(t,s) \geq c(a,b)\Phi(s) \text{ for } t\in [a,b] \text{ and }
s \in [0,1].
\end{gather*}
\end{itemize}
This is often proved by showing the following condition for
some $c \in P\setminus\{0\}$,
\begin{equation} \label{eq-c}
c(t) \Phi(s) \leq G(t,s) \leq \Phi(s), \quad
\text{for}\; 0 \leq t,\; s \leq 1.
\end{equation}
If \eqref{eq-c} is valid we can take any $[a,b] \subset [0,1]$
for which $c(a,b):=\min_{t \in [a,b]}c(t)>0$. When $(C)$ holds,
$S$ maps $P$ into the sub-cone $K$ where
\begin{equation}
\label{eq-K}
K:=\{u \in P: u(t) \ge c(a,b)\|u\| \text{ for } t \in [a,b]\}.
\end{equation}
We also use some comparisons with the associated linear operator
\begin{equation}\label{eq-linop}
Lu(t):=\int_0^1 G(t,s)g(s)u(s)\,ds.
\end{equation}
Under the above conditions $L(P) \subset K$. It is known that,
under the above conditions, $S,L$ are completely continuous and
that the spectral radius of $L$, denoted $r(L)$, is positive. By
the Krein-Rutman theorem, $r(L)$ is an eigenvalue of $L$ with an
eigenfunction in $P$, hence also in $K$; $r(L)$ is usually called
the principal eigenvalue of $L$. We let $\mu(L):=1/r(L)$ be the
principal characteristic value of $L$.
The existence result in \cite{jw-gi-jlms-06} uses some constants
defined as follows.
\begin{equation} \label{eq-mM}
\begin{gathered}
m:=\Bigl(\,\sup_{t\in [0,1]}\int_{0}^{1}
G(t,s)g(s)\,ds\Bigr)^{-1}, \\
M=M(a,b):=\Bigl(\,\inf_{t\in
[a,b]}\int_{a}^{b}G(t,s)g(s)\,ds\Bigr)^{-1}.
\end{gathered}
\end{equation}
We use the following notations.
\begin{gather*}
f^{0}=\limsup_{u \to 0+}f(u)/u, \quad
f_{0}=\liminf_{u \to 0+}f(u)/u;\\
f^{\infty}=\limsup_{u \to \infty}f(u)/u, \quad
f_{\infty}=\liminf_{u \to \infty}f(u)/u.
\end{gather*}
The existence theorem for one positive solution reads as follows.
\begin{theorem}[\cite{jw-gi-jlms-06}] \label{thm-exist1}
Suppose {\rm (C)} holds for every $[a,b] \subset(0,1)$. Then equation
\eqref{eq-nonintop} has a positive solution $u\in K$ if one of the
following conditions holds.
\begin{itemize}
\item[$(S_1)$] $0\le f^{0}<\mu(L)$ and ${\mu(L)}0$
then we must impose the sublinear or superlinear conditions on $f$.
The result for at least two positive solutions is the following.
\begin{theorem}[\cite{jw-gi-jlms-06}]\label{thm-exist2}
Suppose {\rm (C)} holds for some $[a,b] \subset [0,1]$.
Then equation \eqref{eq-nonintop} has at least two positive
solutions in $K$ if one of the following holds.
\begin{itemize}
\item[$(D_1)$] $0\leq f^{0}<\mu(L)$, there is $\rho>0$ such that
$f(u)>M\rho$ for all $u \in [\rho, \rho/c(a,b)]$, and
$0\le f^{\infty}<\mu(L)$.
\item[$(D_2)$] $\mu(L)0$, $f(u)< m\rho$ for all $u \in [0,\rho]$, and
${\mu(L)}M$ for all
$u \in [\rho, \rho/c(a,b)]$'' is equivalent to ``$f(u)>c(a,b)M$
for all $u \in [c(a,b)\rho_1, \rho_1]$''.
This condition is less restrictive when $c(a,b)$ is as large as
possible, the length of the interval on which the condition holds
is less, and when $M(a,b)$ is as small as possible,
the height to be exceeded by $f(u)$ is less. However, a minimal
$M$ could correspond to a very small $c$ so the actual choice
of $[a,b]$ depends on properties of the given nonlinearity.
We have found that aiming to make $M(a,b)$ as small as possible
is often a good choice.
\end{remark}
There is another recent result that complements $(D_1)$ and
applies when we have the following condition to replace (C).
\begin{itemize}
\item[$(C_0)$] There exist $\Phi_1(s)$ and a constant $c_0>0$ such that
$c_0 \Phi_1(s) \le G(t,s) \le \Phi_1(s)$, for all
$t,s \in [0,1]$.
\end{itemize}
\begin{theorem}[\cite{jw-jlms-10}] \label{thm-exist2bis}
Suppose $(C_0)$ holds. Then equation \eqref{eq-nonintop} has at
least two positive solutions in $K$ if
\begin{itemize}
\item[$(D'_1)$] $0\leq f^{0}<{\mu(L)}$, there exists $\rho>0$
such that $f(u)>{\mu(L)} u$
for $\rho \leq u \leq \rho/c_0 $, and $0\le f^{\infty}<{\mu(L)}$.
\end{itemize}
\end{theorem}
The reason why this complements $(D_1)$ is the following.
It is known that $m \le \mu(L) \le M(a,b)$ for every
$[a,b] \subset [0,1]$ (\cite{jw-kql-tmna}), and, if $c(a,b)$
and $c_0$ have been found as large as possible then $c_0 \le c(a,b)$.
At the point $\rho$ the condition $f(\rho)>\mu(L) \rho$
is weaker than $f(u) > M(a,b)\rho$, but at the point $u=\rho/c(a,b)$,
the condition $f(u)>\mu(L)u$ can be more restrictive than
$f(u)>M(a,b)\rho$, depending on the values of $M(a,b)$
and $c(a,b)$. It is easy to give examples where either one
of the conditions is applicable but the other is not.
There are also non-existence results
see \cite{jw-jlms-10,jw-gi-jlms-09}, which show that the hypotheses
in the Theorem \ref{thm-exist1} are sharp.
\begin{theorem}\label{thm-nonex}
The operator $S$ defined in \eqref{eq-nonintop} has no nonzero
fixed points in $P$ if either \begin{equation*}
f(t,u)<{\mu}(L)u \quad \text{for all } u > 0,
\end{equation*}
or
\begin{equation*}
f(t,u)>{\mu}(L)u \quad \text{for all } u > 0.
\end{equation*}
\end{theorem}
\section{The Green's function and related properties}
We now consider the BVP
\begin{equation} \label{eq31}
u''(t)+{\lambda}g(t)f(u(t))=0, \quad t \in(0,1),
\end{equation}
with the nonlocal BCs
\begin{equation}
\label{bc31}
u(0)=\beta[u]:=\int_0^1 u(s)\,dB(s), \quad
u(1)=k\beta[u]=k\int_0^1 u(s)\,dB(s), \quad (k>0),
\end{equation}
where $dB$ is a Stieltjes measure, that is, $B$ in non-decreasing.
A typical example is when
\begin{equation*}
\beta[u]=\sum_{i=1}^{m} \beta_{i}u(\xi_i)+\int_0^1 b(s)u(s)\,ds.
\end{equation*}
with $\xi_i \in (0,1)$, under positivity conditions on $\beta_i$
and $b$.
This is a more general version of the problem studied
in \cite{C-T-ejde2011}.
We will determine the Green's function and show how the
conditions $(C)$ and $(C_0)$ can be established. These results are new.
The Green's function can be found from the explicit formulae
in \cite{jw-gi-jlms-06}, or, for those who prefer to work
with matrices, from \cite{jw-gi-jlms-09}. However,
there is a shortcut, which uses the \emph{methods} that were
used to obtain these formulae, which we now give.
Let $\gamma_1(t):=1-t, \gamma_2(t):=t$. The Green's function for
the BVP
\begin{equation} \label{eq-3bvp}
u''(t)+{\lambda}g(t)f(u(t))=0, \;t \in(0,1), \quad
u(0)=0, u(1)=0,
\end{equation}
is well-known and can be written
\begin{equation} \label{eq-gf0}
G_0(t,s)=\begin{cases}
\gamma_1(t)\gamma_2(s), & s\le t,\\
\gamma_1(s)\gamma_1(t), & s>t.
\end{cases}
\end{equation}
Hence it follows readily that
\begin{equation}
\label{eq-c03}
c_0(t)\Phi_0(s) \le G_0(t,s) \le \Phi_0(s), \quad
\text{for all } s,t \in [0,1],
\end{equation}
where $\Phi(s)=s(1-s)$, $c_0(t)=\min\{t,1-t\}$.
We write
\begin{equation*}
S_0u(t)=\int_0^1 G_0(t,s) g(s)f(u(s))\,ds.
\end{equation*}
It is now easy to see that solutions of the BVP \eqref{eq-3bvp}
are fixed points of the nonlinear operator
\begin{equation*}
Tu(t):=\gamma_1(t)\beta[u]+\gamma_2(t)k\beta[u] +S_0u(t)
={\gamma(t)}\beta[u]+S_0u(t),
\end{equation*}
where $\gamma(t)=1+(k-1)t$.
Hence, if $\beta[\gamma] \neq 1$, by applying the functional
$\beta$ and then replacing $\beta[u]$
(for details see any of
\cite{jw-gi-nodea08,jw-gi-jlms-06,jw-gi-jlms-09}
in increasing degrees of generality), fixed points of $T$
are fixed points of $S$ where
\begin{equation*}
Su(t)=\dfrac{\gamma(t)}{1-\beta[\gamma]}\int_0^1 \mathcal{G}(s)g(s)f(u(s))\,ds+\int_0^1 G_{0}(t,s) g(s)f(u(s))\,ds,
\end{equation*}
where
\begin{equation} \label{eq-calg}
\mathcal{G}(s):=\int_0^1 G_0(t,s)\,dB(t).
\end{equation}
Thus, the Green's function for the BVP \eqref{eq-3bvp} is given by
\begin{equation} \label{eq-gf3}
G(t,s)=\dfrac{\gamma(t)}{1-\beta[\gamma]}\,\mathcal{G}(s)+G_0(t,s).
\end{equation}
We will suppose that $0 \le \beta[\gamma]<1$ in order that $S$
maps $P$ into $P$. When sign changing measures are used it
is also required that $\mathcal{G}(s) \ge 0$; for (positive)
measures that we are now considering this holds automatically.
The expression \eqref{eq-gf3} can be checked by a longer
calculation using the formulae in \cite{jw-gi-jlms-06}
or \cite{jw-gi-jlms-09}. The authors of \cite{C-T-ejde2011}
found their expression for the Green's function in a different
form by a direct, longer, calculation. The form we find has
several advantages: it is easily determined; it applies
to many BCs at once; the properties required follow easily from
those of $G_0$, which are simpler to discuss,
for example positivity of $G$ is now obvious.
Let $\gamma_{m}:=\min_{t \in [0,1]}\gamma(t)=\min\{1,k\}$.
Clearly we have
\begin{gather}
\label{eq-3comp}
G(t,s) \le \dfrac{\|\gamma\|}{1-\beta[\gamma]}\,\mathcal{G}(s)
+\Phi_0(s),\\
G(t,s) \ge \dfrac{\gamma_{m}}{1-\beta[\gamma]}\,\mathcal{G}(s)
+c_0(t)\Phi_0(s).
\end{gather}
Thus if we let $\Phi(s):=\frac{\|\gamma\|}{1-\beta[\gamma]}\,
\mathcal{G}(s)+\Phi_0(s)$
we have
\begin{equation*}
G(t,s)\ge \min\bigl\{\frac{m_{\gamma}}{\|\gamma\|},
c_0(t)\bigr\}\Phi(s).
\end{equation*}
Thus condition $(C)$ holds with
$c(t)=\min\bigl\{\frac{m_{\gamma}}{\|\gamma\|}, c_0(t)\bigr\}$
which in this case is
\begin{equation}\label{eq-cc}
c(t)=\min\bigl\{\dfrac{\min\{1,k\}}{\max\{1,k\}}, t, 1-t\bigr\}.
\end{equation}
We will also show that $(C_0)$ holds for the problem
\eqref{eq31}, \eqref{bc31}.
We note that, since $c_0(t)\Phi_0(s) \le G_0(t,s)$, we have
\begin{equation} \label{eq-ineq}
\mathcal{G}(s)=\int_0^1 G_0(t,s)\,dB(t) \ge \Phi_0(s)\int_0^1 c_0(t)\,dB(t)=\beta[c_0]\Phi_0(s).
\end{equation}
Therefore, from \eqref{eq-3comp}, we obtain
\begin{equation*}
G(t,s) \le \dfrac{\|\gamma\|}{1-\beta[\gamma]}\,
\mathcal{G}(s)+\dfrac{1}{\beta[c_0]}\mathcal{G}(s).
\end{equation*}
Hence, taking $\Phi_1(s):=\dfrac{\|\gamma\|}{1-\beta[\gamma]}\,\mathcal{G}(s)
+\dfrac{1}{\beta[c_0]}\,\mathcal{G}(s)$,
we have
\begin{equation*}
G(t,s) \ge \dfrac{\gamma_{m}}{1-\beta[\gamma]}\,\mathcal{G}(s)
\ge \dfrac{\gamma_{m}}{\|\gamma\|+\frac{1-\beta[\gamma]}{\beta[c_0]}}\,\Phi_1(s).
\end{equation*}
Thus, $(C_0)$ holds with
\begin{equation}\label{eq-czero}
c_0= \frac{\gamma_{m}}{\|\gamma\|+\frac{1-\beta[\gamma]}{\beta[c_0]}}.
\end{equation}
\section{Examples}
The authors of \cite{C-T-ejde2011} studied the problem
with BCs
\begin{equation}
\label{bc41}
u(0)=\beta_1\int_0^\eta u(s)\,ds, \quad
u(1)=\beta_2\int_0^\eta u(s)\,ds.
\end{equation}
The given conditions on the coefficients $\beta_1, \beta_2$ are
\begin{equation*}
0 < \beta_2\eta^2 <2, \quad\text{and} \quad
0 < \beta_1<\dfrac{2-\beta_2{\eta}^2}{\eta(1-\eta/2)},
\end{equation*}
which is equivalent to the condition
$0 < \beta[\gamma]=\beta_2\eta^2/2+\beta_1(\eta-\eta^2/2)<1$,
in our notation.
By concavity arguments, they essentially proved
(\cite[Lemma 2.4]{C-T-ejde2011}) a result equivalent to
$G$ satisfying condition $(C_0)$ with a constant
\begin{equation}
\label{ct-const}
\widetilde{c}_0=\min\Bigl\{\dfrac{\beta_2\eta(1-\eta)}{2-\beta_1\eta-\beta_2\eta^2},
\dfrac{\beta_2\eta^2}{2-\beta_1\eta},\dfrac{\beta_1\eta(1-\eta)}{2-\beta_1\eta},
\dfrac{\beta_1\eta^2}{2-\beta_1\eta}\Bigr\}.
\end{equation}
We shall see that, in the first example we give, this is smaller
than the constant found in \eqref{eq-czero}, hence would always
give a worse result than we could give.
\begin{example} \label{ex2} \rm
We first consider Example 4.3 from \cite{C-T-ejde2011}.
We have given several numbers rounded to five decimal places
but all numbers in this example can be given with greater accuracy.
Let $f(u):=u^2e^{-u}$. The problem is
\begin{equation}\label{eq-ex2}
\begin{gathered}
u''(t)+{\lambda} f(u(t)), \quad 01/4.
\end{cases}
\end{equation*}
We now determine the constants required in Theorem \ref{thm-exist2}.
We choose $[a,b]$ so as to minimize $M(a,b)$. We note that
$c(t)=\min\{m_{\gamma}/{\|\gamma\|}, c_0(t)\}=\min\{1/10, t,1-t\}$
and we can have $c(t)=1/10$ for all intervals
$[a,b]\subset [1/10,9/10]$.
We used Maple to calculate $M(a,b)$ on such intervals and
calculated $M(0.273, 0.9)\approx 2.70608$ which is close
to minimal for such intervals.
We remark that $f(u)$ is increasing for $0 \le u \le 2$ and is
decreasing for $u \ge 2$; also $f(u)/u$ is increasing for $0 __M\rho$ for $\rho \le u \le \rho/c(a,b)=10\rho$.
If $10 \rho \le 2$ then by the above remarks we must choose $\rho$
so that ${\lambda}f(\rho)>M\rho$, that is,
${\lambda}>M\exp(\rho)/\rho$ and the least $\lambda$ is obtained
by choosing $\rho=1/5$. Thus, for $\rho=1/5$ we find $\lambda
f(u)>\rho M$ for $\rho \le u \le \rho/c(a,b)=10\rho$ if
$\lambda>5M\exp(1/5) \approx 16.52606$.
If $\rho>2$ we must have ${\lambda}f(10\rho)>M\rho$, that is,
${\lambda}100{\rho}^2\exp(-10\rho)>M \rho$, or
$\lambda>\dfrac{M}{10}\dfrac{\exp(10\rho)}{10\rho}$ and the least
possibility is $\lambda>\dfrac{M}{10}\dfrac{\exp(20)}{20} \approx
6.56448 \times 10^6$, thus this is not a good choice, though it
gives a better constant than $5.0536 \times10^{13}$.
If $\rho<2<10\rho$ then we choose $\rho$ so that
$\exp(\rho)/\rho=\exp(10\rho)/(100\rho)$ and find $\rho \approx
0.51169$ and then it suffices to have $\lambda>8.82185$.
We now give a lower estimate using Theorem~\ref{thm-nonex}. For
this we need the principal characteristic value $\mu(L)$. By
considering eigenfunctions of the form $\sin(\omega t+\theta)$ and
using the boundary conditions, we find, using Maple to solve some
equations involving trigonometric functions, that $\mu(L) \approx
1.36469$. If ${\lambda}f(u)<\mu(L)u$ for all $u \ge 0$ there is no
positive solution, thus there is no positive solution if
$\lambda<\exp(1)\mu(L)\approx 3.70963$.
We now see how Theorem~\ref{thm-exist2bis} with condition $(D2')$
applies in this example. We now want to find $r$ such that
$\lambda f(u) \ge \mu(L) u$ for all $r \le u \le r/c_0=13r$. Since
$f(u)/u$ is increasing for $0 ____ \mu(L) r/f(r) \approx 7.90619$. Thus, in
this case, $(D2')$ gives a better result than $(D2)$.
In summary, using the theory of Webb-Infante \cite{jw-gi-jlms-06}
we have shown that
\begin{quote}
for $\lambda>\lambda^{**} \approx 8.82185$ the problem has at
least two positive solutions, and
for $\lambda<\mu^{**}=\exp(1)\mu(L)\approx 3.70963$ there are no
positive solutions.
\end{quote}
Using the newer result from \cite{jw-jlms-10} we can improve
this to get $\lambda^{**} \approx
7.90619$. These estimates of $\lambda^{**}$ are substantial
improvements of that from \cite{C-T-ejde2011} written above.
\begin{example}\label{ex3} \rm
We now consider Example 4.2 from \cite{C-T-ejde2011},
with a changed notation to make the interval $[0,1]$.
Let $f(u):=u^{1/2}+u^2$, let $g(t)=(1-t)^{1/2}$. The problem is
\begin{equation}\label{eq-ex3}
\begin{gathered}
u''(t)+{\lambda}g(t) f(u(t)), \quad 00$, thus there are no positive solutions if
$\lambda> \mu^*=\mu(L) p/f(p) \approx 4.431$.
Again the constant $\lambda^*$ here is a substantial improvement
on the result of \cite{C-T-ejde2011}.
Our discussion shows that it is important in examples to choose
$\rho$ to fit the behaviour of $f$.
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