\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 11, pp. 1--8.\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-2011/11\hfil Existence of positive solutions]
{Existence of positive solutions for self-adjoint
boundary-value problems with integral boundary conditions at
resonance}

\author[A. Yang, B. Sun, W. Ge\hfil EJDE-2011/11\hfilneg]
{Aijun Yang, Bo Sun, Weigao Ge}  % not in alphabetical order

\address{Aijun Yang \newline
College of Science, Zhejiang University of Technology, 
Hangzhou, Zhejiang, 310032, China}
\email{yangaij2004@163.com}

\address{Bo Sun \newline
School of Applied Mathematics, 
Central University of Finance and Economics, 
Beijing, 100081, China}
\email{sunbo19830328@163.com}

\address{Weigao Ge \newline
Department of Applied Mathematics,
Beijing Institute of Technology, 
Beijing, 100081, China}
\email{gew@bit.edu.cn}

\thanks{Submitted September 29, 2010. Published January 20, 2011.}
\thanks{Supported by grant 11071014 from NNSF of China,
by the Youth PhD Development Fund \hfill\break\indent
of CUFE 121 Talent Cultivation Project}
\subjclass[2000]{34B10, 34B15, 34B45}
\keywords{Boundary value problem; resonance; cone;
positive solution; \hfill\break\indent coincidence}

\begin{abstract}
 In this article, we study the self-adjoint second-order
 boundary-value problem with integral boundary conditions,
 \begin{gather*}
 (p(t)x'(t))'+ f(t,x(t))=0,\quad t\in (0,1),\\
 p(0)x'(0)=p(1)x'(1),\quad x(1)=\int_0^1x(s)g(s)ds,
 \end{gather*}
 which involves an integral boundary condition.
 We prove the existence of positive solutions
 using a new tool: the  Leggett-Williams norm-type theorem
 for coincidences.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

This paper concerns the existence of
positive solutions to the following boundary value problem at
resonance:
\begin{gather}
(p(t)x'(t))'+ f(t,x(t))=0,\quad t\in (0,1), \label{e1.1}\\
p(0)x'(0)=p(1)x'(1),\quad x(1)=\int_0^1x(s)g(s)ds,\label{e1.2}
\end{gather}
where $g\in L^1[0,1]$ with $g(t)\geq0$ on $[0,1]$,
$\int_0^1g(s)ds=1$, $p\in C[0,1]\cap C^1(0,1)$, $p(t)>0$ on
$[0,1]$.

 Recently much attention has been paid to the study of
certain nonlocal boundary value problems (BVPs). The methodology for
dealing with such problems varies. For example,  Kosmatov \cite{k1}
applied a coincidence degree theorem due to Mawhin and obtained the
existence of at least one solution of the BVP at resonance
\begin{gather*}
u''(t)=f(t,u(t),u'(t)),~~t\in (0,1),\\
u'(0)=u'(\eta),\quad \sum_{i=1}^{n}\alpha_iu(\eta_i)=u(1),
\end{gather*}
under the assumptions $\sum_{i=1}^{n}\alpha_i=1$ and
$\sum_{i=1}^{n}\alpha_i\eta_i=1$.

Han \cite{h1} studied the three-point BVP at resonance
\begin{gather*}
x''(t)=f(t,x(t)),\quad t\in (0,1),\\
x'(0)=0,\quad x(\eta)=x(1).
\end{gather*}
The author rewrote the original BVP as an equivalent problem, and
then used the Krasnolsel'skii-Gue fixed point theorem.

Although the existing literature on solutions of BVPs is quite wide,
to the best of our knowledge, only a few papers deal with the
existence of positive solutions to multi-point BVPs at resonance.
In particular, there has been no work done for the
BVP \eqref{e1.1}-\eqref{e1.2}. Moreover, Our
main approach is different from the ones existing and our main
ingredient is the Leggett-Williams norm-type theorem for
coincidences obtained by O'Regan and Zima \cite{o1}.

\section{Related Lemmas}

For the convenience of the reader, we review some
standard facts on Fredholm operators and cones in Banach spaces. Let
$X$, $Y$ be real Banach spaces. Consider a linear mapping
$L:\operatorname{dom}L\subset X\to Y$ and a nonlinear operator
$N:X\to Y$. Assume that
\begin{itemize}
\item[(A1)] $L$ is a Fredholm
operator of index zero; that is,  $\operatorname{Im}L$ is closed and\\
$\operatorname{dim}\ker L=\operatorname{codim\, Im}L<\infty$.
\end{itemize}
This assumption implies that there exist continuous projections
$P:X\to X$ and $Q:Y\to Y$ such that
$\operatorname{Im}P=\ker L$ and $\ker Q=\operatorname{Im}L$.
Moreover, since $\text{dim Im}Q=\operatorname{codim\ Im}L$,
there exists an isomorphism $J:\operatorname{Im}Q\to \ker L$.
Denote by $L_p$ the restriction of $L$ to
$\ker P\cap \operatorname{dom}L$.
Clearly, $L_p$ is an isomorphism from
$\ker P\cap \operatorname{dom}L$ to $\operatorname{Im}L$,
we denote its inverse by
$K_p:\operatorname{Im}L\to \ker P\cap \operatorname{dom}L$. It is
known (see \cite{m1}) that the coincidence equation $Lx=Nx$ is equivalent
to
$$
x=(P+JQN)x+K_P(I-Q)Nx.
$$
Let $C$ be a cone in $X$ such that
\begin{itemize}
\item[(i)] $\mu x\in C$ for all $x\in C$ and $\mu\geq 0$,
\item[(ii)] $x,-x\in C$ implies $x=\theta$.
\end{itemize}
It is well known that $C$ induces a partial order in $X$ by
$$
x\preceq y \quad \text{if and only if}\quad  y-x\in C.
$$

 The following property is valid for every cone in a Banach
 space $X$.

\begin{lemma}[\cite{p1}] \label{lem2.1}
 Let $C$ be a cone in $X$. Then for
every $u\in C\setminus\{0\}$ there exists a positive number
$\sigma(u)$ such that
$$
\|x+u\|\geq \sigma(u)\|u\| \quad\text{for all }x\in C.
$$
\end{lemma}

 Let $\gamma:X\to C$ be a retraction; that is, a
continuous mapping such that $\gamma (x)=x$ for all $x\in C$. Set
$$
\Psi:=P+JQN+K_p(I-Q)N\quad \text{and}\quad
\Psi_\gamma:=\Psi\circ\gamma.
$$
We  use  the following result due to O'Regan and Zima, with
the following assumptions:
\begin{itemize}
\item[(A2)]
$QN:X\to Y$ is continuous and bounded and
$K_p(I-Q)N:X\to X$ be compact on every bounded subset of
$X$,
\item[(A3)] $Lx\neq \lambda Nx$ for all $x\in
C\cap\partial\Omega_2\cap \emph{Im}L$ and $\lambda\in (0,1)$,

\item[(A4)] $\gamma$ maps subsets of $\overline{\Omega}_2$ into
 bounded subsets of $C$,

\item[(A5)]  $ \deg\{[I-(P+JQN)\gamma]|_{\ker L},
\ker L\cap\Omega_2,0\}\neq 0$,

\item[(A6)] there exists $u_0\in C\setminus\{0\}$ such that
$\|x\|\leq \sigma(u_0)\|\Psi x\|$ for
$x\in C(u_0)\cap\partial\Omega_1$, where $C(u_0)=\{x\in C:\mu
u_0\preceq x ~for~some ~\mu>0 \}$ and $\sigma(u_0)$ such that
$\|x+u_0\|\geq \sigma(u_0)\|x\|$ for every $x\in C$,

\item[(A7)] $(P+JQN)\gamma(\partial\Omega_2)\subset C$,

\item[(A8)]
$\Psi_\gamma(\overline{\Omega}_2\setminus\Omega_1)\subset C$.
\end{itemize}

\begin{theorem}[\cite{o1}] \label{thm2.1}
 Let $C$ be a cone in $X$ and let
$\Omega_1$, $\Omega_2$ be open bounded subsets of $X$ with
$\overline{\Omega}_1\subset\Omega_2$ and
$C\cap(\overline{\Omega}_2\setminus\Omega_1)\neq \emptyset$. Assume
that {\rm (A1)--(A8)} hold.
Then the equation $Lx=Nx$ has a solution in the set
$C\cap(\overline{\Omega}_2\setminus\Omega_1)$.
\end{theorem}

 For simplicity of notation, we set
\begin{equation}
\begin{gathered}
\omega:=\int_0^1(\int_{s}^1\frac{1}{p(\tau)}d\tau)g(s)ds,\\
l(s):=\int_{s}^1\Big(\int_{\tau}^1\frac{1}{p(r)}dr\Big)g(\tau)d\tau
+ \int_{s}^1\frac{1}{p(\tau)}d\tau\int_0^{s}g(\tau)d\tau,
\end{gathered}\label{e2.1}
\end{equation}
and
\[
G(t,s)
=\begin{cases}
\frac{1}{\omega}\big[\int_0^{s}(\int_{s}^1\frac{1}{p(r)}dr-\int_{\tau}^1\frac{r}{p(r)}dr)g(\tau)d\tau
+\int_{s}^1\int_{\tau}^1\frac{1-r}{p(r)}drg(\tau)d\tau\big]
\\
\times\big[\int_0^1\frac{\tau}{p(\tau)}d\tau
-\int_{t}^1\frac{1}{p(\tau)}d\tau\big]
+1+\int_0^1\frac{\tau^{2}}{p(\tau)}d\tau
+\int_{t}^1\frac{1-\tau}{p(\tau)}d\tau
-\int_{s}^1\frac{\tau}{p(\tau)}d\tau,\\
\quad \text{if }0\leq s < t\leq1,\\[4pt]
\frac{1}{\omega}\big[\int_0^{s}(\int_{s}^1\frac{1}{p(r)}dr
-\int_{\tau}^1\frac{r}{p(r)}dr)g(\tau)d\tau
+\int_{s}^1\int_{\tau}^1\frac{1-r}{p(r)}drg(\tau)d\tau\big]\\
\times\big[\int_0^1\frac{\tau}{p(\tau)}d\tau
 -\int_{t}^1\frac{1}{p(\tau)}d\tau\big]
+1+\int_0^1\frac{\tau^{2}}{p(\tau)}d\tau
+\int_{s}^1\frac{1-\tau}{p(\tau)}d\tau
-\int_{t}^1\frac{\tau}{p(\tau)}d\tau ,\\
\quad \text{if }0\leq t\leq s\leq1.
\end{cases}
\]
Note that $G(t,s)\geq0$ for $t,s\in [0,1]$, and set
\begin{equation}
\kappa:=\min\big\{1,\;\frac{1}{\max_{t,s\in[0,1]}G(t,s)}\big\}.
\label{e2.2}
\end{equation}

\section{Main result}

To prove the existence result, we present here a definition.

\begin{definition} \label{def3.1} \rm
 We say that the function $f:[0,1]\times\mathbb{R}\to
 \mathbb{R}$ satisfies the $L^1$-Carath\'eodory conditions,
 if
\begin{itemize}
\item[(i)] for each $u\in\mathbb{R}$, the mapping $t\mapsto
 f(t,u)$ is Lebesgue measurable on $[0,1]$,
\item[(ii)]  for a.e. $t\in [0,1]$, the mapping $u\mapsto
 f(t,u)$ is continuous on $\mathbb{R}$,
\item[(iii)] for each $r>0$, there exists $\alpha_{r}\in L^1[0,1]$
 satisfying $\alpha_{r}(t)>0$ on $ [0,1]$ such that
 $$
 |u|\leq r\text{ implies }|f(t,u)|\leq  \alpha_{r}(t).
 $$
\end{itemize}
\end{definition}

Now, we state our result on the existence of positive solutions
for \eqref{e1.1}-\eqref{e1.2}. under the following assumptions:
\begin{itemize}
\item[(H1)] $f:[0,1]\times \mathbb{R}\to \mathbb{R}$
satisfies the $L^1$-Carath\'eodory conditions,

\item[(H2)] there exist positive constants $b_1, b_2, b_3, c_1,c_2,
B$ with
\begin{equation}
B>\frac{c_2}{c_1}+3(\frac{b_2c_2}{b_1c_1}+\frac{b_3}{b_1})
\int_0^1\frac{1+s}{p(s)}ds, \label{e3.1}
\end{equation}
such that
\[
-\kappa x\leq f(t,x),\quad
f(t,x)\leq -c_1x+c_2,\quad
f(t,x)\leq -b_1|f(t,x)|+b_2x+b_3
\]
for $t\in[0,1]$, $x\in[0,B]$,

\item[(H3)] there exist $b\in (0,B)$, $t_0\in [0,1]$,
$\rho\in (0,1]$, $\delta\in (0,1)$ and $q\in L^1[0,1]$,
$q(t)\geq 0$ on $[0,1]$, $h\in C([0,1]\times(0,b],\mathbb{R}^{+})$
such that $f(t,x)\geq q(t)h(t,x)$ for $t\in[0,1]$ and $x\in(0,b]$.
For each $t\in[0,1]$, $\frac{h(t,x)}{x^{\rho}}$ is non-increasing on
$x\in(0,b]$ with
\begin{equation}
\int_0^1G(t_0,s)q(s)\frac{h(s,b)}{b}ds\geq
\frac{1-\delta}{\delta^{\rho}}.\label{e3.2}
\end{equation}
\end{itemize}

\begin{theorem} \label{thm3.1}  Under assumptions {\rm (H1)--(H3)},
The problem \eqref{e1.1}-\eqref{e1.2} has at least one positive
 solution on $[0,1]$.
\end{theorem}

\begin{proof}
Consider the Banach spaces $X=C[0,1]$ with the supremum norm
$\|x\|=\max_{t\in[0,1]}|x(t)|$ and  $Y=L^1[0,1]$ with the
usual integral norm  $\|y\|=\int_0^1|y(t)|dt$.
Define $L: \operatorname{dom}L\subset X\to Y$ and $N:X\to Y$ with
\begin{align*}
\operatorname{dom}L=\big\{&x\in X:
p(0)x'(0)=p(1)x'(1),\; x(1)=\int_0^1x(s)g(s)ds,\\\
& x,px'\in AC[0,1],\; (px')'\in L^1[0,1]\big\}
\end{align*}
with $Lx(t)=-(p(t)x'(t))'$ and $Nx(t)=f(t,x(t))$, $t\in[0,1]$.
Then
\begin{gather*}
\ker L=\{x\in \operatorname{dom}L: x(t)\equiv c\text{ on }[0,1]\},\\
\operatorname{Im}L=\{y\in Y:\int_0^1y(s)ds=0\}.
\end{gather*}
 Next, we define the projections
$P:X\to X$ by $(Px)(t)=\int_0^1x(s)ds$ and
$Q:Y\to Y$ by
$$
(Qy)(t)=\int _0^1y(s)ds.
$$
Clearly, $\operatorname{Im}P= \ker L$ and
$\ker Q=\operatorname{Im}L$. So
$\operatorname{dim\,ker}L=1=\operatorname{dim\, Im}Q
=\operatorname{codim\, Im}L$.
 Notice that $\operatorname{Im}L$ is closed, $L$
is a Fredholm operator of index zero; i.e. (A1) holds.

Note that the inverse
$K_p:\operatorname{Im}L\to \operatorname{dom}L\cap
\ker P$ of $L_p$ is given by
$$
(K_py)(t)=\int_0^1k(t,s)y(s)ds,
$$
where
\begin{equation}
k(t,s):=\begin{cases}
-\int_{s}^1\frac{\tau}{p(\tau)}d\tau+\frac{1}{\omega}l(s)
\big[\int_0^1\frac{\tau}{p(\tau)}d\tau
-\int_{t}^1\frac{1}{p(\tau)}d\tau\big]\\
+\int_{t}^1\frac{1}{p(\tau)}d\tau,
 &0\leq s\leq t\leq1,\\[4pt]
-\int_{s}^1\frac{\tau}{p(\tau)}d\tau+\frac{1}{\omega}l(s)
\big[\int_0^1\frac{\tau}{p(\tau)}d\tau
-\int_{t}^1\frac{1}{p(\tau)}d\tau\big]\\
+\int_{s}^1\frac{1}{p(\tau)}d\tau,
 &0\leq t< s\leq1,
\end{cases} \label{e3.3}
\end{equation}
It is easy to see that
$|k(t,s)|\leq 3\int_0^1\frac{1+s}{p(s)}ds$.
Since $f$ satisfies the $L^1$-Carath\'eodory
conditions, (A2) holds.

Consider the cone
$$
C=\{x\in X: x(t)\geq 0\text{ on }[0,1]\}.
$$
 Let
\begin{gather*}
\Omega_1=\{x\in X: \delta\|x\|<|x(t)|<b\text{ on }[0,1]\},\\
\Omega_2=\{x\in X:\|x\|<B\}.
\end{gather*}
Clearly, $\Omega_1$ and $\Omega_2$ are bounded and open sets and
$$
\overline{\Omega}_1=\{x\in X:
\delta\|x\|\leq|x(t)|\leq b\text{ on }[0,1]\}\subset\Omega_2
$$
(see \cite{o1}). Moreover,
$C\cap(\overline{\Omega}_2\setminus\Omega_1)\neq\emptyset$. Let
$J=I$ and $(\gamma x)(t)=|x(t)|$ for $x\in X$. Then $\gamma$ is a
retraction and maps subsets of $\overline{\Omega}_2$ into bounded
subsets of $C$, which means that 4$^{\circ}$ holds.

To prove (A3), suppose that there exist $x_0\in
\partial\Omega_2\cap C\cap\operatorname{dom}L $ and
$\lambda_0\in(0,1)$ such that $Lx_0=\lambda_0Nx_0$, then
$(p(t)x_0'(t))'+\lambda_0f(t,x_0(t))=0$ for all $t\in[0,1]$. In view
of (H2), we have
$$
-\frac{1}{\lambda_0}(p(t)x'_0(t))'=f(t,x_0(t))
\leq -\frac{1}{\lambda_0}b_1|(p(t)x'_0(t))'|+b_2x_0(t)+b_3.
$$
Hence,
$$
0\leq -b_1\int_0^1|(p(t)x'_0(t))'|dt
 +\lambda_0b_2\int_0^1x_0(t)dt+\lambda_0b_3,
$$
which gives
\begin{equation}
\int_0^1|(p(t)x'_0(t))'|dt\leq
\frac{b_2}{b_1}\int_0^1x_0(t)dt+\frac{b_3}{b_1}.\label{e3.4}
\end{equation}
Similarly, from (H2), we also obtain
\begin{equation}
\int_0^1x_0(t)dt\leq \frac{c_2}{c_1}.\label{e3.5}
\end{equation}
On the other hand,
\begin{equation}
\begin{aligned}
x_0(t)
&= \int_0^1x_0(t)dt+\int_0^1k(t,s)(p(s)x'_0(s))'ds\\
&\leq \int_0^1x_0(t)dt+\int_0^1|k(t,s)|\,
|(p(s)x'_0(s))'|ds.
\end{aligned}\label{e3.6}
\end{equation}
 From \eqref{e3.4}, \eqref{e3.5} and \eqref{e3.6}, we have
$$
B=\|x_0\|\leq \frac{c_2}{c_1}+3(\frac{b_2c_2}{b_1c_1}
+\frac{b_3}{b_1})\int_0^1\frac{1+s}{p(s)}ds,
$$
which contradicts (H2).

To prove (A5), consider $x\in \ker L\cap\overline{\Omega}_2$.
Then $x(t)\equiv c$ on $[0,1]$. Let
$$
H(c,\lambda)=c-\lambda |c|-\lambda\int_0^1 f(s,|c|)ds
$$
for $c\in[-B,B]$ and $\lambda\in[0,1]$. It is
easy to show that $0= H(c,\lambda)$ implies $c\geq 0$.
 Suppose $0=H(B,\lambda)$ for some $\lambda\in(0,1]$. Then,
(H2) leads to
$$
0\leq B(1-\lambda)=\lambda\int
_0^1f(s,B)ds\leq\lambda(-c_1B+c_2)<0
$$
which is a contradiction.
In addition, if $\lambda=0$, then $B=0$, which is impossible.
Thus, $H(x,\lambda)\neq 0$ for
$x\in \ker L\cap\partial{\Omega}_2$,
$\lambda\in[0,1]$. As a result,
$$
\deg\{H(\cdot,1), \ker L\cap\Omega_2,0\}
=\deg\{H(\cdot,0), \ker L\cap\Omega_2,0\}.
$$
However,
$$
\deg\{H(\cdot,0), \ker L\cap\Omega_2,0\}
=\deg\{I, \ker L\cap\Omega_2,0\}=1.
$$
Then
$$
\deg\{[I-(P+JQN)\gamma]_{\ker L}, \ker L\cap\Omega_2,0\}
=\deg\{H(\cdot,1), \ker L\cap\Omega_2,0\}\neq0.
$$

Next, we prove (A8). Let
$x\in\overline{\Omega}_2\setminus\Omega_1$ and $t\in[0,1]$,
\begin{align*}
(\Psi_\gamma
x)(t)&=\int_0^1|x(s)|ds+\int_0^1f(s,|x(s)|)ds\\
 &\quad+\int_0^1k(t,s)[f(s,|x(s)|)-\int_0^1f(\tau,|x(\tau)|)d\tau]ds\\
&= \int_0^1|x(s)|ds+\int_0^1G(t,s)f(s,|x(s)|)ds\\
&\geq \int_0^1(1-\kappa G(t,s))|x(s)|ds\geq 0.
\end{align*}
 Hence, $\Psi_\gamma(\overline{\Omega}_2\setminus\Omega_1)\subset C$;
 i.e. (A8) holds.

Since for $x\in\partial\Omega_2$,
\begin{align*}
(P+JQN)\gamma x&= \int_0^1|x(s)|ds+\int_0^1f(s,|x(s)|)ds\\
&\geq \int_0^1(1-\kappa)|x(s)|ds\geq0.
\end{align*}
Thus, $(P+JQN)\gamma x\subset C$ for $x\in\partial\Omega_2$,
(A7) holds.

 It remains to verify (A6). Let
$u_0(t)\equiv1$ on $[0,1]$. Then $u_0\in C\setminus\{0\}$,
$C(u_0)=\{x\in C:x(t)>0\text{ on }[0,1]\}$
 and we can take
$\sigma(u_0)=1$. Let $x\in C(u_0)\cap \partial\Omega_1$. Then
$x(t)>0$ on $[0,1]$, $0<\|x\|\leq b$ and $x(t)\geq \delta \|x\|$ on
$[0,1]$. For every $x\in C(u_0)\cap \partial\Omega_1$, by (H3), we
have
\begin{align*}
(\Psi x)(t_0)
&= \int_0^1x(s)ds+\int_0^1G(t_0,s)f(s,x(s))ds\\
&\geq \delta\|x\|+\int_0^1G(t_0,s)q(s)h(s,x(s))ds\\
&= \delta\|x\|+\int_0^1G(t_0,s)q(s)\frac{h(s,x(s))}{x^{\rho}(s)}
 x^{\rho}(s)ds\\
&\geq \delta\|x\|+\delta^{\rho}\|x\|^{\rho}\int_0^1G(t_0,s)
 q(s)\frac{h(s,b)}{b^{\rho}}ds\\
&= \delta\|x\|+\delta^{\rho}\|x\|\cdot
 \frac{b^{1-\rho}}{\|x\|^{1-\rho}}\int_0^1G(t_0,s)q(s)
 \frac{h(s,b)}{b}ds\\
&\geq \delta\|x\|+\delta^{\rho}\|x\|\int_0^1G(t_0,s)q(s)
 \frac{h(s,b)}{b}ds
\geq \|x\|.
\end{align*}
Thus, $\|x\|\leq\sigma(u_0)\|\Psi x\|$ for all $x\in C(u_0)\cap
\partial\Omega_1$.

By Theorem \ref{thm2.1}, the BVP \eqref{e1.1}-\eqref{e1.2} has a positive
solution $x^{*}$ on $[0,1]$ with $b\leq\|x^{*}\|\leq B$.
This completes the proof.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
Note that with the projection $P(x)=x(0)$,
conditions (A7) and (A8) of Theorem \ref{thm2.1} are no longer
satisfied.
\end{remark}

 To illustrate how our main result can be used in
practice, we present here an example.

\subsection*{Example}
Consider the problem
\begin{equation}
\begin{gathered}
(e^{54t}(1+t)x'(t))'+f(t,x(t))=0,\quad t\in (0,1),\\
x'(0)=2e^{54}x'(1),\quad x(1)=\int_0^12sx(s)ds.
\end{gathered} \label{e3.7}
\end{equation}
Corresponding to  \eqref{e1.1}-\eqref{e1.2}, we have
\begin{gather*}
p(t)=e^{54t}(1+t),\quad g(t)=2t, \\
f(t,x)=\begin{cases}
\sin (\pi x/2), &(t,x)\in [0,1]\times(-\infty,3),\\
2-x, &(t,x)\in [0,1]\times[3,+\infty).
\end{cases}
\end{gather*}
When $\kappa=1/2$, choose $c_1=1$, $c_2=3$,
$b_1=1/2$, $b_2=3/2$, $b_3=9/2$, $B=4$ and
$b=1/2$, $t_0=0$, $\rho=1$, $\delta=1/2$,
$q(t)=1-t$, $h(t,x)=\sin (\pi x/2)$. We can check that all
the conditions of Theorem \ref{thm3.1} are satisfied,
 then the BVP \eqref{e3.7} has a positive solution on $[0,1]$.

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\section*{Addendum posted on March 14, 2011}

In response to comments from a reader, we want to make the
following corrections:

Page 2, Line 9: Delete the last sentence in the introduction:
``Moreover, \dots by O'Regan and Zima \cite{o1}".
Then insert the following paragraph:

Using the Legget-Williams norm-type theorem for coincidences, 
which is a tool introduced by O'Regan and Zima \cite{o1}, 
Infante and Zima \cite{i1} studied the multi-point boundary-value problem 
\begin{gather*}
x''(t)=f(t,x(t))=0,\\
x'0)=0, \quad x(1)=\sum_{i=1}^{m-2} \alpha_i x(\eta_i)\,.
\end{gather*}
Inspired by the work in \cite{i1,o1}, we follow their steps,
use the Legget-Williams norm-type theorem, and
quote some of their results.
\medskip

Page 6, Line $-3$: Replace $b\leq \|x^*\|\leq B$ by
$\|x^*\|\leq B$.
\medskip

The authors want to thank the anonymous reader for the suggestions. 


\end{document}