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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 106, 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  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/106\hfil Positive solutions]
{Positive solutions for multipoint boundary-value problem with parameters}

\author[J. Xu,  Z. Wei\hfil EJDE-2008/106\hfilneg]
{Juanjuan Xu, Zhongli Wei}

\address{Juanjuan Xu \newline
School of Mathematics, Shandong University, 
Jinan, Shandong 250100, China}
\email{jnxujuanjuan@163.com Tel: 86-531-88369649}

\address{Zhongli Wei \newline
School of Mathematics, Shandong University, 
Jinan, Shandong 250100, China} 
\email{jnwzl@yahoo.com.cn}

\thanks{Submitted May 2, 2008. Published August 7, 2008.}
\thanks{Supported by grants 10771117 from the National Natural Science
Foundation of China, \hfill\break\indent
and 306001 from the Foundation of School of
Mathematics, Shandong University}
\subjclass[2000]{34B15, 39A10}
\keywords{Multipoint; positive solution;  eigenvalue; parameters}

\begin{abstract}
 In this paper, we study a  generalized Sturm-Liouville
 boundary-value problems with two positive parameters. By
 constructing a completely continuous operator and combining fixed
 point index theorem and some properties of the eigenvalues of linear
 operators, we obtain sufficient conditions for the existence of at
 least one positive solution.
\end{abstract}

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

\section{Introduction}

Multipoint boundary-value problems for ordinary differential
equations arise in different areas of applied mathematics and
physics. For example, the vibrations of a guy wire of uniform
cross-section and composed of N parts of different densities can
be set up as a multipoint boundary-value problem; many problem in
the theory of elastic stability can be handled as multipoint
boundary-value problems too. Recently, the existence and
multiplicity of positive solutions for nonlinear ordinary
differential equations have received a great deal of attention. To
identify a few cases, we refer the readers to
\cite{m1,z1,z2,z3} and references therein.

Li \cite{l1} studied the following boundary-value problem (BVP for
short):
\begin{equation}
\begin{gathered}
u^{(4)}(t)+\beta u''-\alpha u=f(t,u(t)), \quad 0<t<1,\\
u(0)=u(1)=u''(0)=u''(1)=0,
\end{gathered}\label{e1.1}
\end{equation}
where the function $f\in C([0,1]\times [0,+\infty),[0,+\infty)),\
\alpha,\beta\in\mathbb{R}$ and satisfy $\beta<2\pi^{2},\
\alpha\geq-\frac{\beta^{2}}{4},\
\frac{\alpha}{\pi^{4}}+\frac{\beta}{\pi^{2}}<1$. By applications
of the fixed point index theory, sufficient conditions for
existence of at least one positive solution are established.

Ma \cite{m2} studied the existence of positive solution for BVP:
\begin{equation} \label{e1.2}
 \begin{gathered}
 u^{(4)}(t)+\alpha u''-\beta u=f(t,u(t)), \quad 0<t<1,\\
 u(0)=\sum_{i=1}^{m-2}\alpha_iu(\xi_i),\quad
u(1)=\sum_{i=1}^{m-2}\beta_iu(\xi_i),\\
u''(0)=\sum_{i=1}^{m-2}\alpha_iu''(\xi_i),\quad
u''(1)=\sum_{i=1}^{m-2}\beta_iu''(\xi_i),
\end{gathered}
\end{equation}
where $\alpha,\beta\in\mathbb{R}$ and $\alpha<2\pi^{2}$,
$\beta\geq-\frac{\alpha^{2}}{4}$,
$\alpha_i,\beta_i,\xi_i>0$ ($i=1,2,\ldots,m-2$) are constants, and
$f\in C([0,1]\times [0,+\infty),[0,+\infty))$. The main tool is
also the fixed point index theory.

Motivated by the results mentioned above, we are concerned with
the existence of at least one positive solution for the following
generalized Sturm-Liouville BVP:
\begin{equation} \label{e1.3}
\begin{gathered}
u^{(4)}(t)-\beta u''+\alpha u= f(t,u(t)), \quad 0<t<1,\\
au(0)-bu'(0)=\sum_{i=1}^{m-2}\alpha_iu(\xi_i),\quad
cu(1)+du'(1)=\sum_{i=1}^{m-2}\beta_iu(\xi_i),\\
au''(0)-bu'''(0)=\sum_{i=1}^{m-2}\alpha_iu''(\xi_i),\quad
cu''(1)+du'''(1)=\sum_{i=1}^{m-2}\beta_iu''(\xi_i),
\end{gathered}
\end{equation}
where $f\in C([0,1]\times[0,+\infty),[0,+\infty))$ satisfying
$f(t,u)\not\equiv0$ and $\alpha,\beta\geq0$,
$a,b,c,d\in[0,+\infty)$
and $\rho:=ac+bc+ad>0$, $\xi_{i}\in(0,1)$,
$\alpha_{i},\beta_{i}\in[0,+\infty)$ ($i=1,2,\dots,m-2$) are
constants.

To study \eqref{e1.3}, we set up an integral equation which is
equivalent to \eqref{e1.3}. By using the classical fixed point index
theorem and combining some knowledge about eigenvalue of linear
operator, we obtain a sufficient condition for the existence of at
least one positive solution.

Following theorems are needed.

\begin{theorem}[\cite{g2}] \label{thm1.1}
Let $E$ be a Banach space, and let
$P\subset E$ be a cone. Assume $\Omega(P)$ is a bounded open set
in $P$. Suppose that $A:\overline{\Omega(P)}\to P$ is a
completely continuous operator. If there exists $\psi_{0}\in
P\backslash \{\theta\}$ such that
$\varphi-A\varphi\neq\mu\psi_{0}$, for all $\varphi\in\partial\Omega(P)$,
$ \mu\geq0$, then the fixed point index satisfies
$i(A,\Omega(P),P)=0$.
\end{theorem}

 \begin{theorem}[\cite{g2}] \label{thm1.2}
Let $E$ be a Banach space, and let $P\subset E$ be a cone.
 Assume $\Omega(P)$ is a
bounded open set in $P$ with $\theta\in\Omega(P)$. Suppose that
$A:\overline{\Omega(P)}\to P$ is a completely continuous
operator. If $A\psi\neq\mu\psi$, for all $\psi\in\partial\Omega(P)$,
$\mu\geq1$, then the fixed point index satisfies
$i(A,\Omega(P),P)=1$.
\end{theorem}

We shall organize this paper as follows. In Section 2, we present
some preliminaries and lemmas for use later. Finally, we obtain
the main result and state the proof. \vspace{0.2cm}

\section{Preliminaries}

In this section, we state some useful preliminary results and
change the BVP \eqref{e1.3} into the fixed point problem in a cone.
First, we state the following hypothesis to assumed in this paper.
\begin{itemize}
\item[(H1)] $\alpha,\beta\geq0$ and $\alpha\leq \beta^{2}/4$.
\end{itemize}

\begin{remark} \label{rmk2.1} \rm
 From {\rm (H1)}, it follows  that
$\frac{\alpha}{\pi^{4}}+\frac{\beta}{\pi^{2}}>-1$.
\end{remark}

\begin{lemma} \label{lem2.1}
 Under assumption {\rm (H1)} there exist unique
$\varphi_{1},\varphi_{2},\psi_{1},\psi_{2}$ satisfying
 \begin{gather*}
-\varphi_{i}''(t)+\lambda_{i}\varphi_{i}=0, \quad 0<t<1,\\
\varphi_{i}(0)=b,\quad \varphi_{i}'(0)=a, \\
-\psi_{i}''(t)+\lambda_{i}\psi_{i}=0, \quad 0<t<1,\\
\psi_{i}(1)=d,\quad \psi_{i}'(1)=-c,
\end{gather*}
for $i=1,2$.  Also on $[0,1]$,
$\varphi_{1},\varphi_{2},\psi_{1},\psi_{2}\geq0$, where
$\lambda_{1},\lambda_{2}$ are the roots for the polynomial equation
$\lambda^{2}-\beta\lambda+\alpha=0$;
i.e.,
$$
\lambda_{1}=\frac{\beta+\sqrt{\beta^{2}-4\alpha}}{2},
\quad
\lambda_{2}=\frac{\beta-\sqrt{\beta^{2}-4\alpha}}{2}.
$$
Moreover, $\varphi_{1},\varphi_{2}$ are nondecreasing on $[0,1]$ and
$\psi_{1},\psi_{2}$ are nonincreasing on $[0,1]$.
\end{lemma}

\begin{proof}
 From (H1), we have $\lambda_{1},\lambda_{2}\geq0$. By computations
we get that:
If $\lambda_{i}>0$, then
$\varphi_{i}(t)=b\cosh\sqrt{\lambda_{i}}t+\frac{a}{\sqrt{\lambda_{i}}}
\sinh\sqrt{\lambda_{i}}t$,
$$
\psi_{i}(t)=d\cosh\sqrt{\lambda_{i}}(1-t)+\frac{c}{\sqrt{\lambda_{i}}}\sinh
\sqrt{\lambda_{i}}(1-t),
\quad (i=1,2);
$$
if $\lambda_{i}=0$, then $\varphi_{i}(t)=b+at$,
$\psi_{i}(t)=d+c-ct$, ($i=1,2$).

It is obvious that on $[0,1]$,
$\varphi_{1},\varphi_{2},\psi_{1},\psi_{2}\geq0$ and
$\varphi_{1},\varphi_{2}$ are nondecreasing on $[0,1]$,
$\psi_{1},\psi_{2}$ are nonincreasing on $[0,1]$.
\end{proof}

 We denote
$$
\rho_{1}=\left|\begin{matrix}
\psi_{1}(0)  & \varphi_{1}(0) \\
\psi_{1}'(0) & \varphi_{1}'(0)
\end{matrix}\right|, \quad
\rho_{2}=\left|\begin{matrix}
\psi_{2}(0) &\varphi_{2}(0) \\
\psi_{2}'(0) &\varphi_{2}'(0)
\end{matrix}\right|,
$$
\begin{gather*}
 \Delta_{1}=\left|\begin{matrix}
-\sum_{i=1}^{m-2}\alpha_i\varphi_{1}(\xi_i)
&\rho_{1}-\sum_{i=1}^{m-2}\alpha_i\psi_{1}(\xi_i)
\\
\rho_{1}-\sum_{i=1}^{m-2}\beta_i\varphi_{1}(\xi_i)
&-\sum_{i=1}^{m-2}\beta_i\psi_{1}(\xi_i)
\end{matrix} \right|,\\
\Delta_{2}=\left|\begin{matrix}
-\sum_{i=1}^{m-2}\alpha_i\varphi_{2}(\xi_i)
&\rho_{2}-\sum_{i=1}^{m-2}\alpha_i\psi_{2}(\xi_i) \\
\rho_{2}-\sum_{i=1}^{m-2}\beta_i\varphi_{2}(\xi_i)
&-\sum_{i=1}^{m-2}\beta_i\psi_{2}(\xi_i)
\end{matrix}\right|\,.
\end{gather*}
Assume that
\begin{itemize}

\item[(H2)] $\Delta_{1}<0$,
$\rho_{1}-\sum_{i=1}^{m-2}\alpha_i\psi_{1}(\xi_i)>0$,
$\rho_{1}-\sum_{i=1}^{m-2}\beta_i\varphi_{1}(\xi_i)>0$;

\item[(H3)] $\Delta_{2}<0$,
$\rho_{2}-\sum_{i=1}^{m-2}\alpha_i\psi_{2}(\xi_i)>0$,
$\rho_{2}-\sum_{i=1}^{m-2}\beta_i\varphi_{2}(\xi_i)>0$,
\end{itemize}
Similar to \cite{m4}, we can get the following two lemmas by direct
calculations.

\begin{lemma} \label{lem2.2}
 Let {\rm (H1)-(H2)} hold. Then for any
$g\in  C[0,1]$, the  problem
\begin{equation} \label{e2.1}
\begin{gathered}
-u''(t)+\lambda_{1}u(t)=g(t), \quad 0<t<1,\\
 au(0)-bu'(0)=\sum_{i=1}^{m-2}\alpha_iu(\xi_i),\quad
cu(1)+du'(1)=\sum_{i=1}^{m-2}\beta_iu(\xi_i),
\end{gathered}
\end{equation}
has a unique solution
$u(t)=\int_0^1G_{1}(t,s)g(s)\, ds
+A_{1}(g)\varphi_{1}(t)+B_{1}(g)\psi_{1}(t)$,
 where
\begin{gather*}
G_{1}(t,s)=\frac{1}{\rho_{1}}\begin{cases}
 \varphi_{1}(t)\psi_{1}(s), & 0\leq t\leq s\leq 1,\\
 \varphi_{1}(s)\psi_{1}(t), & 0\leq s\leq t\leq 1,
\end{cases}
\\
A_{1}(g):=\frac{1}{\Delta_{1}}\left|\begin{matrix}
\sum_{i=1}^{m-2}\alpha_i\int_0^1G_{1}(\xi_{i},s)g(s)\,ds
&\rho_{1}-\sum_{i=1}^{m-2}\alpha_i\psi_{1}(\xi_i) \\
\sum_{i=1}^{m-2}\beta_i\int_0^1G_{1}(\xi_{i},s)g(s)\,ds
&-\sum_{i=1}^{m-2}\beta_i\psi_{1}(\xi_i)
\end{matrix}\right|,
\\
B_{1}(g):=\frac{1}{\Delta_{1}} \left|\begin{matrix}
-\sum_{i=1}^{m-2}\alpha_i\varphi_{1}(\xi_{i})
&\sum_{i=1}^{m-2}\alpha_i\int_0^1G_{1}(\xi_{i},s)g(s)\,
ds \\
\rho_{1}-\sum_{i=1}^{m-2}\beta_i\varphi_{1}(\xi_{i})
&\sum_{i=1}^{m-2}\beta_i\int_0^1G_{1}(\xi_{i},s)g(s)\, ds
\end{matrix}\right|,
\end{gather*}
and where $g\geq0$, $u(t)\geq 0$, $t\in[0,1]$.
\end{lemma}

The proof of the above lemma follows by routine
calculations.

\begin{lemma} \label{lem2.3}
Let {\rm (H1), (H3)} hold. Then for each
$g\in  C[0,1]$, the  problem
\begin{equation} \label{e22.}
\begin{gathered}
-u''(t)+\lambda_{2}u(t)=g(t), \quad 0<t<1,\\
 au(0)-bu'(0)=\sum_{i=1}^{m-2}\alpha_iu(\xi_i),\quad
cu(1)+du'(1)=\sum_{i=1}^{m-2}\beta_iu(\xi_i),
\end{gathered}
\end{equation}
 has a unique solution
$u(t)=\int_0^1G_{2}(t,s)g(s)\,
ds+A_{2}(g)\varphi_{2}(t)+B_{2}(g)\psi_{2}(t)$,
where
\begin{gather*}
G_{2}(t,s)=\frac{1}{\rho_{2}}\begin{cases}
\varphi_{2}(t)\psi_{2}(s),&  0\leq t\leq s\leq 1,\\
\varphi_{2}(s)\psi_{2}(t), & 0\leq s\leq t\leq 1,
\end{cases}
\\
A_{2}(g):=\frac{1}{\Delta_{2}}
\left|\begin{matrix}
\sum_{i=1}^{m-2}\alpha_i\int_0^1G_{2}(\xi_{i},s)g(s)\,
ds  & \rho_{2}-\sum_{i=1}^{m-2}\alpha_i\psi_{2}(\xi_i) \\
\sum_{i=1}^{m-2}\beta_i\int_0^1G_{2}(\xi_{i},s)g(s)\,ds &
-\sum_{i=1}^{m-2}\beta_i\psi_{2}(\xi_i)
\end{matrix}\right|,
\\
B_{2}(g):=\frac{1}{\Delta_{2}}\left|\begin{matrix}
-\sum_{i=1}^{m-2}\alpha_i\varphi_{2}(\xi_{i})
&\sum_{i=1}^{m-2}\alpha_i\int_0^1G_{2}(\xi_{i},s)g(s)\,ds \\
\rho_{2}-\sum_{i=1}^{m-2}\beta_i\varphi_{2}(\xi_{i})
&\sum_{i=1}^{m-2}\beta_i\int_0^1G_{2}(\xi_{i},s)g(s)\,ds
\end{matrix}\right|,
\end{gather*}
and  $g\geq0$, $u(t)\geq0$, $t\in[0,1]$.
\end{lemma}

The proof of the above lemma follows by routine calculations.

\begin{remark} \label{rmk2.2} \rm
Suppose that (H2) and (H3) hold. It follows that
$A_i(g), B_i(g)$ ($i=1,2$) are increasing.
\end{remark}

\begin{lemma} \label{lem2.4}
 Assume that {\rm (H1)--(H3)} hold. Then
\eqref{e1.3} has a unique solution
\begin{equation} \label{e2.3}
\begin{aligned}
u(t)&=\int_0^1\int_0^1G_{2}(t,\tau)G_{1}(\tau,s)f(s,u(s))\,ds\,
d\tau+\int_0^1G_{2}(t,\tau)A_{1}(f)\varphi_{1}(\tau)\, d\tau\\
&\quad +\int_0^1G_{2}(t,\tau)B_{1}(f)\psi_{1}(\tau)\, d\tau
+A_{2}(h)\varphi_{2}(t)+B_{2}(h)\psi_{2}(t),
\end{aligned}
\end{equation}
where $G_{1},G_{2},A_{1},A_{2},B_{1},B_{2}$ are defined as above,
$$
h(t)=\int_0^1G_{1}(t,s)f(s,u(s))\,
ds+A_{1}(f)\varphi_{1}(t)+B_{1}(f)\psi_{1}(t).
$$
\end{lemma}

Obviously, $u(t)\geq0$ for all $t\in[0,1]$. Let $E=C[0,1]$ and
$P=\{u\in E,u\geq0\}$. It is obvious that $P$ is a cone in $E$.
Define $T:E\to E$,
\begin{equation} \label{e2.4}
\begin{aligned}
 Tu(t)&=\int_0^1\int_0^1G_{2}(t,\tau)G_{1}(\tau,s)f(s,u(s))\,ds\,d\tau
+\int_0^1G_{2}(t,\tau)A_{1}(f)\varphi_{1}(\tau)\, d\tau\\
 &\quad +\int_0^1G_{2}(t,\tau)B_{1}(f)\psi_{1}(\tau)\, d\tau
+A_{2}(h)\varphi_{2}(t)+B_{2}(h)\psi_{2}(t),
\end{aligned}
\end{equation}
where $h(t)=\int_0^1G_{1}(t,s)f(s,u(s))\,ds
+A_{1}(f)\varphi_{1}(t)+B_{1}(f)\psi_{1}(t)$.

We can easily obtain that $u$ is a positive solution of \eqref{e1.3}
if and only if $u$ is a fixed point of $T$ in $P$.

Define $L:E\to E$,
\begin{equation} \label{e2.5}
\begin{aligned}
Lu(t)&=\int_0^1\int_0^1G_{2}(t,\tau)G_{1}(\tau,s)u(s)\,ds\,d\tau
+\int_0^1G_{2}(t,\tau)A_{1}(u)\varphi_{1}(\tau)\,d\tau\\
&\quad +\int_0^1G_{2}(t,\tau)B_{1}(u)\psi_{1}(\tau)\, d\tau
+A_{2}(e)\varphi_{2}(t)+B_{2}(e)\psi_{2}(t),
\end{aligned}
\end{equation}
where $e(t)=\int_0^1G_{1}(t,s)u(s)\,
ds+A_{1}(u)\varphi_{1}(t)+B_{1}(u)\psi_{1}(t)$.

\begin{lemma} \label{lem2.5}
Suppose that {\rm (H1)--(H3)} hold. Then $T:P\to P$ is completely
continuous. Also $L:P\to P$ is completely continuous.
\end{lemma}

\begin{lemma} \label{lem2.7}
 Suppose that {\rm (H1)--(H3)} hold. Then for
the operator $L$ defined by \eqref{e2.5}, the spectral radius $r(L)\neq0$
and $L$ has a positive eigenfunction corresponding to its first
eigenvalue $\lambda_{*}=r(L)^{-1}$.
\end{lemma}

\begin{proof}
It is easy to see that there is $t_{1}\in(0,1)$, such
that $G_{1}(t_{1},t_{1})G_{2}(t_{1},t_{1})>0$. Thus there exists
$[\alpha,\beta]\subset(0,1)$ such that $t_{1}\in(\alpha,\beta)$ and
$G_{1}(t,\tau)G_{2}(\tau,s)>0,\ t,\tau,s\in[\alpha,\beta]$.

Take $u\in E$ such that $u(t)\geq0$ for all $t\in[0,1]$,
$u(t_{1})>0$ and $u(t)=0$ for all
$t\in[0,1]\backslash[\alpha,\beta]$. Then for $t\in [\alpha,\beta]$,
\begin{align*}
 Lu(t)&=\int_0^1\int_0^1G_{2}(t,\tau)G_{1}(\tau,s)u(s)\,ds\,d\tau
+\int_0^1G_{2}(t,\tau)A_{1}(u)\varphi_{1}(\tau)\,d\tau\\
&\quad +\int_0^1G_{2}(t,\tau)B_{1}(u)\psi_{1}(\tau)\, d\tau
+A_{2}(e)\varphi_{2}(t)+B_{2}(e)\psi_{2}(t)\\
&\geq\int_\alpha^\beta\int_\alpha^\beta
G_{2}(t,\tau)G_{1}(\tau,s)u(s)\, ds\,d\tau
+\int_\alpha^\beta G_{2}(t,\tau)A_{1}(u)\varphi_{1}(\tau)\,d\tau\\
&\quad +\int_\alpha^\beta
G_{2}(t,\tau)B_{1}(u)\psi_{1}(\tau)\, d\tau
+A_{2}(e)\varphi_{2}(t)+B_{2}(e)\psi_{2}(t)
>0.
\end{align*}
So there exists a constant $c>0$ such that for $t\in[0,1]$,
$c(Lu)(t)\geq u(t)$. From Krein-Rutmann Theorem \cite{g2}, we know
that the spectral radius $r(L)\neq0$ and $L$ has a positive
eigenfunction corresponding to its first eigenvalue
$\lambda_{*}=r(L)^{-1}$.
\end{proof}

\section{Main Result}

\begin{theorem} \label{thm3.1}
Suppose that {\rm (H1)--(H3)} hold, and
$\underline{f_{0}}>\lambda_{*}, \overline{f_{\infty}}<\lambda_{*}$,
where $\lambda_{*}$ is the first eigenvalue of $L$ defined by \eqref{e2.5}.
Then  \eqref{e1.3} has at least one positive solution, where
$$
\underline{f_{0}}=\liminf_{u\to 0^{+}}\min_{t\in[0,1]}\frac{f(t,u)}{u},\quad
\overline{f_{\infty}}=\limsup_{u\to+\infty}\max_{t\in[0,1]}\frac{f(t,u)}{u}.
$$
\end{theorem}

\begin{proof}
 From $\underline{f_{0}}>\lambda_{*}$, there exists
$r_{1}>0$, such that
 $f(t,u)\geq\lambda_{*} u$ for all $t\in[0,1]$,
$u\in [0,r_{1}]$.
 Let $u\in\partial B_{r_{1}}\cap P$. Then
\begin{align*}
 Tu(t)&=\int_0^1\int_0^1G_{2}(t,\tau)G_{1}(\tau,s)f(s,u(s))\,ds\,d\tau
+\int_0^1G_{2}(t,\tau)A_{1}(f)\varphi_{1}(\tau)\,d\tau\\
&\quad +\int_0^1G_{2}(t,\tau)B_{1}(f)\psi_{1}(\tau)\, d\tau
+A_{2}(h)\varphi_{2}(t)+B_{2}(h)\psi_{2}(t)\\
&\geq \lambda_{*}[\int_0^1\int_0^1G_{2}(t,\tau)G_{1}(\tau,s)u(s)\,ds\,
d\tau
+\int_0^1G_{2}(t,\tau)A_{1}(u)\varphi_{1}(\tau)\,d\tau\\
&\quad +\int_0^1G_{2}(t,\tau)B_{1}(u)\psi_{1}(\tau)\, d\tau
+A_{2}(e)\varphi_{2}(t)+B_{2}(e)\psi_{2}(t)]\\
&=\lambda_{*}(Lu)(t).
\end{align*}

We may suppose that $T$ has no fixed point on $\partial
B_{r_{1}}\cap P$(otherwise, the proof is complete). Now we show that
$u-Tu\neq\mu u^{*}$ for all $u\in\partial B_{r_{1}}\cap P$,
$\mu\geq0$.

Otherwise, there exists $u_{1}\in\partial B_{r_{1}}\cap P$,
$\tau_{0}\geq0$, such that $u_{1}-Tu_{1}=\tau_{0}u^{*}$, that is
$$
u_{1}=Tu_{1}+\tau_{0}u^{*}.
$$
Let $\tau^{*}=\sup\{\tau:u_{1}\geq \tau u^{*}\}$, then
$\tau^{*}\geq\tau_{0}>0$, and $u_{1}\geq \tau^{*} u^{*}$. Since
$L(P)\subset P$,
$\lambda_{*}Lu_{1}\geq\tau^{*}\lambda_{*}Lu^{*}=\tau^{*}u^{*}$, we
have
$$
u_{1}=Tu_{1}+\tau_{0}u^{*}\geq\lambda_{*}Lu_{1}+\tau_{0}u^{*}
\geq(\tau^{*}+\tau_{0})u^{*}.
$$
which contradicts the definition of $\tau^{*}$, so
$i(T,B_{r_{1}}\cap P,P)=0$.

From $\overline{f_{\infty}}<\lambda_{*}$, there exits $0<\sigma<1$,
$r_{2}>r_{1}$, such that $f(t,u)\leq\sigma\lambda_{*}u$ for all
$t\in[0,1]$, $u\in[r_{2},+\infty)$. Let
$L_{1}u=\sigma\lambda_{*}Lu$, $u\in E$, then $L_{1}:E\to E$ is a
bounded linear operator and $L_{1}(P)\subset P$. Let
\begin{align*}
M^{*}
&=\max_{u\in \overline{B}_{r_{2}}\cap P,t\in[0,1]}
 \int_0^1\int_0^1G_{2}(t,\tau)G_{1}(\tau,s)f(s,u(s))\,ds\,d\tau\\
&\quad +\int_0^1G_{2}(t,\tau)A_{1}(f)\varphi_{1}(\tau)\,d\tau
 + \int_0^1G_{2}(t,\tau)B_{1}(f)\psi_{1}(\tau)\, d\tau\\
&\quad +A_{2}(h)\varphi_{2}(t)+B_{2}(h)\psi_{2}(t),
\end{align*}
obviously, $0<M^{*}<+\infty$. Let
$W=\{u\in P:u=\mu Tu,\;0\leq\mu\leq 1\}$, for all $u\in W$, denote
$\widehat{u(t)}=\min\{u(t),r_{2}\},\;
s(u)=\{t\in[0,1],u(t)>r_{2}\}$,
$\widehat{f(t)}=f(t,\widehat{u(t)})$.
Then
\begin{align*}
u(t)&=\mu Tu(t)\leq Tu(t)\\
&=\int_0^1\int_{s(u)}G_{2}(t,\tau)G_{1}(\tau,s)f(s,u(s))\,ds\,
d\tau
+\int_0^1G_{2}(t,\tau){A_{1}}_{s(u)}(f)\varphi_{1}(\tau)\,d\tau\\
&\quad +\int_0^1G_{2}(t,\tau){B_{1}}_{s(u)}(f)\psi_{1}(\tau)\, d\tau
+A_{2}(h_{s(u)})\varphi_{2}(t)+B_{2}(h_{s(u)})\psi_{2}(t)\\
&\quad
+\int_0^1\int_{[0,1]/s(u)}G_{2}(t,\tau)G_{1}(\tau,s)f(s,u(s))\,
ds\,d\tau \\
&\quad +\int_0^1G_{2}(t,\tau){A_{1}}_{[0,1]/s(u)}(f)\varphi_{1}(\tau)\,d\tau\\
&\quad
+\int_0^1G_{2}(t,\tau){B_{1}}_{[0,1]/s(u)}(f)\psi_{1}(\tau)\,d\tau\\
&\quad
+A_{2}(h_{[0,1]/s(u)})\varphi_{2}(t)+B_{2}(h_{[0,1]/s(u)})\psi_{2}(t)
\\
&\leq\sigma\lambda_{*}[\int_0^1\int_0^1G_{2}(t,\tau)G_{1}(\tau,s)u(s)\,ds\,
d\tau
+\int_0^1G_{2}(t,\tau)A_{1}(u)\varphi_{1}(\tau)\,d\tau \\
&\quad +\int_0^1G_{2}(t,\tau)B_{1}(u)\psi_{1}(\tau)\, d\tau
+A_{2}(e)\varphi_{2}(t)+B_{2}(e)\psi_{2}(t)]\\
&\quad +\int_0^1\int_0^1G_{2}(t,\tau)G_{1}(\tau,s)f(s,\widehat{u(s)})\,
ds\,d\tau
+\int_0^1G_{2}(t,\tau)A_{1}(\widehat{f})\varphi_{1}(\tau)\,d\tau\\
&\quad +\int_0^1G_{2}(t,\tau)B_{1}(\widehat{f})\psi_{1}(\tau)\, d\tau
+A_{2}(\widehat{f})\varphi_{2}(t)+B_{2}(\widehat{f})\psi_{2}(t)\\
&\leq(L_{1}u)(t)+M^{*},\quad t\in[0,1].
\end{align*}
where
\[
{A_{1}}_{s(u)}(f):=\frac{1}{\Delta_{1}}
\left|\begin{matrix}
\sum_{i=1}^{m-2}\alpha_i\int_{s(u)}G_{1}(\xi_{i},s)g(s)\,ds
&\rho_{1}-\sum_{i=1}^{m-2}\alpha_i\psi_{1}(\xi_i) \\
\sum_{i=1}^{m-2}\beta_i\int_{s(u)}G_{1}(\xi_{i},s)g(s)\,ds
&-\sum_{i=1}^{m-2}\beta_i\psi_{1}(\xi_i)
\end{matrix}\right|,
\]
${B_{1}}_{s(u)}$, ${A_{2}}_{[0,1]/s(u)}$, ${B_{2}}_{[0,1]/s(u)}$
have the similar meaning and
$$
h_{s(u)}(t)=\int_{s(u)}G_{1}(t,s)f(s,u(s))\,
ds+{A_{1}}_{s(u)}(f)\varphi_{1}(t)+{B_{1}}_{s(u)}(f)\varphi_{2}(t).
$$
Thus
$$
(I-L_{1})u\leq M^{*},\quad t\in[0,1].
$$
Since $u^{*}=\lambda_{*}(Lu^{*})$ and $0<\sigma<1$, we have
$r(L_{1})^{-1}>1$; i.e., $(I-L_{1})^{-1}$ exists and
$$
(I-L_{1})^{-1}=I+L_{1} +L_{1}^{2}+\dots+L_{1}^{n}+\dots.
$$
It follows from $L_{1}(P)\subset P$ that $(I-L_{1})^{-1}(P)\subset P$.
Therefore,
$u(t)\leq (I-L_{1})^{-1}M^{*}$, $t\in[0,1]$, and  $W$ is bounded.
We denote by $\sup W$ the bound of $W$.

Select $r_{3}>\max\{r_{2},\sup W\}$, then for all
$u\in\partial B_{r_{3}}\cap P$, $u\neq\mu Tu$, $0\leq\mu\leq1$;
that is,
$$
Tu\neq\frac{1}{\mu}u,\quad
\frac{1}{\mu}\geq1, \quad \forall u\in\partial B_{r_{3}}\cap P,
$$
so  from Theorem \ref{thm1.2}, we have $i(T, B_{r_{3}}\cap P,P)=1$.
Therefore,
$$
i(T, (B_{r_{3}}\cap P)\backslash(\overline{B}_{r_{1}}\cap P),P)
=i(T, B_{r_{3}}\cap P,P)-i(T, B_{r_{1}}\cap P,P)=1.
$$
By the solution properties of the fixed point index,  $T$
has at least one fixed point on
$(B_{r_{3}}\cap P)\backslash(\overline{B}_{r_{1}}\cap P)$,
which means that the
generalized Sturm-Liouville boundary-value problem \eqref{e1.3} has at
least one positive solution.
\end{proof}

\subsection*{Acknowledgements}
 The authors would like to thank the anonymous referees for their kind
 suggestions and comments on this paper.

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\end{document}