\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 171, 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 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/171\hfil Positive solutions]
{Positive solutions for singular  Sturm-Liouville boundary value
problems on the half line}

\author[J. Xu, Z. Yang\hfil EJDE-2010/171\hfilneg]
{Jiafa Xu, Zhilin Yang}  % in alphabetical order

\address{
Department of Mathematics,
Qingdao Technological University,
Qingdao, Shandong Province, China}
\email[Jiafa Xu]{xujiafa292@sina.com}
\email[Zhilin Yang]{zhilinyang@sina.com}

\thanks{Submitted April 1, 2010. Published November 30, 2010.}
\thanks{Supported by 10871116 and 10971179 from the
NNSF of China, and ZR2009AL014 from \hfill\break\indent
 the NSF of Shandong Province of China}
\subjclass[2000]{34B40, 34B16, 47H07, 47H11, 45M20}
\keywords{Sturm-Liouville problem on the half line; positive
solution; \hfill\break\indent
fixed point index; spectral radius}

\begin{abstract}
 This article concerns the existence and multiplicity of positive
 solutions for the singular Sturm-Liouville boundary value
 problem
 \begin{gather*}
 (p(t)u'(t))'+h(t)f(t,u(t))=0,\quad 0<t<\infty,\\
 au(0)-b\lim_{t\to 0^+}p(t)u'(t)=0,\\
 c\lim_{t\to \infty}u(t)+d\lim_{t\to \infty}p(t)u'(t)=0.
 \end{gather*}
 We use fixed point index theory to establish our main results based
 on a priori estimates derived by utilizing spectral properties  of
 associated linear integral operators.
\end{abstract}

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

\section{Introduction}

In this article, we study  the singular Sturm-Liouville boundary
value problem on the half line
\begin{equation}\label{problem}
\begin{gathered}
(p(t)u'(t))'+h(t)f(t,u(t))=0,\quad 0<t<\infty,\\
au(0)-b\lim_{t\to 0^+}p(t)u'(t)=0,\\
c\lim_{t\to \infty}u(t)+d\lim_{t\to \infty}p(t)u'(t)=0,
\end{gathered}
\end{equation}
where $f\in C( \mathbb{R}^+\times \mathbb{R}^+, \mathbb{R}^+)$
$(\mathbb{R}^+:=[0,\infty))$, $h$ is nonnegative on $\mathbb{R}^+$
and belongs to a weighted Lebesgue space  on $\mathbb{R}^+$, $p\in
C(\mathbb{R}^+)\cap C^1(0,\infty)$ with $p>0$ on $(0,\infty)$ and
$\int_0^{\infty}\frac{{\rm\,d}s}{p(s)}<\infty$,  $a,b,c,d\geq 0$
with $\rho:=bc+ad+ac\int_0^{\infty}\frac{{\rm\,d}s}{p(s)}>0$.


Boundary value problems on the half line arise  in   studying
radially symmetric  solutions of nonlinear elliptic equations and
 in various applications,  such as an unsteady flow of gas through
 a semi-infinite porous media,
 theory of drain flows, and plasma physics (see for example
\cite{Baxley,Chen,Hai,Tan}). This explains the reason that the last
two decades have witnessed an overgrowing interest in the research
of such  problems, with many papers  in this direction published. We
refer the interested reader to
\cite{Agarwal,Liu1,Liu2,Liu3,Lian1,Lian2,
Ma3,Sun,Tian1,Tian2,Yan,Zhang,Zhang1} and references therein.  In
\cite{Lian2}, by using  fixed point theorems in a cone, Lian et al.
considered problem \eqref{problem} and  obtained  a set of
sufficient conditions that guarantee existence, uniqueness, and
multiplicity of positive solutions for \eqref{problem}. An
interesting feature in \cite{Lian2} is that     the nonlinearity $f$
may  be sign-changing. In \cite{Zhang}, by using fixed point index
theory, Zhang et al. studied the existence of positive solutions for
\eqref{problem} with $h(t)f(t,u(t))$  replaced by the semipositone
nonlinearity $f(t,u(t))+q(t)$, generalizing and improving some
results due to Liu \cite{Liu2} and Zhang et al. \cite{Zhang1}.

 Motivated by the  works cited above,  we discuss the existence
and multiplicity of positive solutions for
\eqref{problem}. We use fixed point index theory to establish our main
results  based on a priori estimates derived by utilizing spectral
properties of associated linear integral operators. This means that
both our methodology and results in this paper
 are  different from those in
\cite{Liu2, Lian1,Lian2,Zhang,Zhang1}.

The article is organized as follows. Section 2 contains some
preliminary results, including  spectral properties of two linear
integral operators. In Section 3, we state and prove our main
results. Four examples are given in Section 4  to illustrate
applications of Theorems \ref{thm1}--\ref{thm3}.

\section{Preliminaries}

Let $E=\{u\in C(\mathbb{R}^+) :\lim_{t\to \infty}u(t) \text{
exists}\}$ be equipped with the supremum  norm  $\|\cdot\|$ and
$P=\{u\in E: u(t)\geq 0, t\in \mathbb{R}^+\}$. Then $(E,\|\cdot\|)$
is  a real Banach space and $P$
 a cone on $E$.   For simplicity, we
denote $\xi(t)$ and $\eta(t)$ by
\[
\xi(t):=b+a\int_0^t \frac{{\rm\,d}s}{p(s)},\quad
\eta(t):=d+c\int_t^{\infty} \frac{{\rm\,d}s}{p(s)}.
\]
Clearly, $c\xi(t)+a\eta(t)=\rho$. Let
\begin{equation}\label{Green}
G(t,s):=\frac{1}{\rho}\begin{cases}
\xi(t)\eta(s),& 0\leq t\leq s<\infty,\\
\xi(s)\eta(t),& 0\leq s\leq t<\infty.\\
\end{cases}
\end{equation}
We  assume  the following conditions hold throughout this article.
\begin{itemize}

\item[(H1)] $h$ is Lebesgue measurable and nonnegative on
$\mathbb{R}^+$, with  $\int_0^{\infty}G(s,s)h(s){\rm\,d}s \in
(0,\infty)$.


\item[(H2)] $p\in C(\mathbb{R}^+)\cap C^1(0,\infty)$, with $p>0$ in
$(0,\infty$) and
 $\int_0^{\infty}\frac{{\rm\,d}s}{p(s)}<\infty$.

\item[(H3)] $f\in C(\mathbb{R}^+\times\mathbb{R}^+, \mathbb{R}^+)$ is
bounded on $\mathbb{R}^+\times [0,R]$ for every $R>0$.
\end{itemize}

\begin{lemma}\label{lemma1}
Let {\rm (H1)-(H3)} hold and $G(t,s)$ be given in \eqref{Green}.
Then \eqref{problem} is equivalent to the fixed point equation
$u=Au$, where $A:P\to P$ is defined by  \begin{equation}\label{A}
(Au)(t):=\int_0^\infty G(t,s)h(s)f(s,u(s)){\rm\,d}s, t\in \mathbb
R^+.\end{equation}
\end{lemma}

\begin{lemma}[\cite{Liu3,Lian2,Zhang}] \label{continous}
 Let  {\rm (H1)-(H3)} hold. Then $A:P\to P$ is a completely
continuous operator.
\end{lemma}

\begin{remark} \label{rmk2.1} \rm
Note that $G$ satisfies  the following properties, which are
obtained from the monotonicity  of $\xi$ and $\eta$:
\begin{itemize}
\item[(1)]  $G(t,s)$ is continuous and $0\leq G(t,s)\leq
G(s,s)$ for all $t,s\in \mathbb{R}^+$,

\item[(2)] $G(t,s)\geq \gamma(t) G(s,s)$ for all $t,s\in
\mathbb{R}^+$, where
\begin{equation}\label{gamma}
\gamma(t):=\min\big\{\frac{\xi(t)}{\xi(\infty)},
\frac{\eta(t)}{\eta(0)}\big\}, \quad t\in\mathbb{R}^+.
\end{equation}
\end{itemize}
\end{remark}

\begin{lemma}[\cite{Guo_1}] \label{lemma2}
Let  $\Omega \subset E$ be  a bounded open set and
$A:\overline{\Omega}\cap P\to P$  a completely continuous
operator. If there exists $u_0\in P\setminus\{0\}$ such that
$u-Au\neq \mu u_0$ for all $\mu \geq 0$ and $u\in
\partial\Omega\cap P$, then $i(A,\Omega\cap P,P)=0$ , where $i$
indicates the fixed point index on $P$.
\end{lemma}

\begin{lemma}[\cite{Guo_1}] \label{lemma3}
 Let $\Omega \subset E$ be a
bounded open set with $0\in \Omega$. Suppose
$A:\overline{\Omega}\cap P\to P$ is a completely continuous
operator. If $u\neq \mu Au$  for all $u\in
\partial\Omega\cap P$ and $0\leq \mu\leq 1$, then
$i(A,\Omega\cap P,P) = 1$.
\end{lemma}

To  establish our main results, we need two extra  completely
continuous linear operators $T$ and $S$, defined by
\begin{equation}
\label{T} (Tu)(t):=\int_0^\infty G(t,s)h(s)u(s){\rm\,d}s,\quad u\in
E,
\end{equation}
\[
(Sv)(s):=\int_0^\infty G(t,s)h(s)v(t){\rm\,d}t,\quad v\in
L(\mathbb{R}^+).
\]
Condition (H1) implies  $S: L(\mathbb{R}^+)\to L(\mathbb{R}^+)$ maps
every nonnegative function in $L(\mathbb{R}^+)$ to a nonnegative
function. Note that $T$  can be viewed as the dual operator of $S$
for the reason  that $T$ can be extended  to a bounded linear
operator $\overline T: L^\infty(\mathbb{R}^+)\to
L^\infty(\mathbb{R}^+)$,  satisfying $\overline
T(L(\mathbb{R}^+))\subset E$. It is easy to  prove
 that the spectral radius of $T$, denoted by $r(T)$,
 is positive.  Now the
well-known Krein-Rutman \cite{Krein} theorem asserts that there
exist a $\varphi\in P\backslash \{0\}$ and a nonnegative $\psi\in
L(\mathbb{R}^+)\setminus\{0\}$  such that $T\varphi=r(T)\varphi$ and
$S\psi=r(T)\psi$, which can be written as
\begin{equation}\label{Krein-Rutman-1}
\int_0^{\infty}
G(t,s)h(s)\varphi(s){\rm\,d}s=r(T)\varphi(t),\end{equation} and
\begin{equation}\label{Krein-Rutman-2}
 \int_0^{\infty} G(t,s)h(s)\psi(t){\rm\,d}t=r(T)\psi(s).
\end{equation}
Note that $\psi$ may be required to satisfy
\begin{equation}\label{integral}
\int_0^{\infty}\psi(t){\rm\,d}t=1.
\end{equation}

\begin{remark} \label{rmk2.2} \rm
 Let $\lambda_1=1/r(T)>0$. Then
\eqref{Krein-Rutman-1} can be written in the form
\begin{equation}\label{eigenvalue}\begin{gathered}
(p(t)\varphi'(t))'+\lambda_1h(t)\varphi(t)=0,\quad 0<t<\infty,\\
a\varphi(0)-b\lim_{t\to 0^+}p(t)\varphi'(t)=0,\\
c\lim_{t\to +\infty}\varphi(t)+d\lim_{t\to
+\infty}p(t)\varphi'(t)=0.
\end{gathered}\end{equation}
This says that $\lambda_1$ is the first eigenvalue  of the above
eigenvalue problem, with  $\varphi$ being a  positive eigenfunction
corresponding to $\lambda_1$.
\end{remark}

 The following is a result that is of crucial importance in our
proofs and,  by Remark 2.3, can be
 proved   as in \cite[Lemma 4]{Yang}.

\begin{lemma}\label{lemma4}
Let  \[ P_0:=\big\{u\in E: \int_0^{\infty}\psi(t)u(t){\rm\,d}t\geq
\omega \|u\|\big\},
\]
where $\psi(t)$ is determined by \eqref{Krein-Rutman-2} and
\eqref{integral}, and $\omega:=\int_0^{\infty}
\gamma(t)\psi(t){\rm\,d}t>0$. Then $T(P)\subset P_0$ and, in
particular,  $\varphi\in P_0$ .
\end{lemma}

\section{Main results}

Let  $B_\delta :=\{u \in E: \|u\| <\delta\}$ for $\delta> 0$.

\begin{theorem}\label{thm1}
Let {\rm (H1)-(H3)} hold. Suppose
\begin{gather}\label{sublinear-1}
\liminf_{u\to 0^+}\frac{f(t,u)}{u}> \lambda_1, \\
\label{sublinear-2}
\limsup_{u\to \infty}\frac{f(t,u)}{u}< \lambda_1
\end{gather}
uniformly for $t\in \mathbb{R}^+$.
Then \eqref{problem} has at least one positive solution.
\end{theorem}

\begin{proof}
By \eqref{sublinear-1}, there exist $r>0$ and $\varepsilon>0$
such that
\[
f(t,u)\geq (\lambda_1+\varepsilon)u,\quad
 \forall t\in \mathbb{R}^+,\; u\in [0,r],
\]
and thus for any  $u\in \overline{B}_r\cap P$, we have
\begin{equation}\label{Inequ-A-1}
(Au)(t)\geq (\lambda_1+\varepsilon)
\int_0^{\infty}G(t,s)h(s)u(s){\rm\,d}s
=(\lambda_1+\varepsilon)(Tu)(t),\quad
t\in \mathbb{R}^+.
\end{equation}
Next  we shall  show that
\begin{equation}\label{notequ-1}
u-Au\neq \mu \varphi,\quad \forall u\in \partial B_r \cap P,
\mu\geq 0,
\end{equation}
where $\varphi$ is defined by \eqref{Krein-Rutman-1}.
If the claim  is false, then there exist $u_1\in
\partial B_r \cap P, \mu_1\ge 0$ such that $u_1-Au_1= \mu_1 \varphi$.
Thus  $u_1=Au_1+\mu_1 \varphi\in P_0$ by Lemma \ref{lemma4} and
$u_1\geq Au_1$. Combining the preceding inequality with
\eqref{Inequ-A-1} (replacing $u$ by $u_1$) leads to
\begin{equation}\label{Inequ-A-2}
u_1(t)\ge (\lambda_1+\varepsilon)
\int_0^{\infty}G(t,s)h(s)u_1(s){\rm\,d}s, t\in\mathbb{R}^+.
\end{equation}
Multiply the above by $\psi(t)$ and integrate over $\mathbb{R}^+$ and
use \eqref{Krein-Rutman-2} and \eqref{integral} to obtain
\[
\int_0^{\infty}u_1(t)\psi(t){\rm\,d}t
\ge\lambda_1^{-1}(\lambda_1+\varepsilon)
\int_0^{\infty}u_1(t)\psi(t){\rm\,d}t,
\]
so that $\int_0^{\infty}u_1(t)\psi(t){\rm\,d}t=0$.
Recalling $u_1\in P_0$,  we  have $u_1\equiv 0$, a contradiction
with  $u_1\in \partial B_r \cap P$. As a result, \eqref{notequ-1} holds. Now
Lemma \ref{lemma2} implies
\begin{equation}\label{index1-0}
i(A,B_r\cap P,P)=0.
\end{equation}
On the other hand, by  \eqref{sublinear-2} and (H3),  there exist
$0<\sigma<1$ and $M>0$ such that
\begin{equation}\label{Inequ-A-3}
f(t,u)\leq \sigma\lambda_1u+M,\quad
\forall u\geq 0,t\in \mathbb{R}^+.
\end{equation}
 We  shall prove that the set
\begin{equation}
\mathcal{M}_1 :=\{u\in P: u=\mu Au, 0\leq \mu\leq 1\}.
\end{equation}
is bounded.  Indeed, for any  $u_2\in\mathcal M_1$ we have
 by \eqref{Inequ-A-3}
\[
u_2(t)\leq \sigma\lambda_1\int_0^{\infty}G(t,s)h(s)u_2(s){\rm\,d}s
+u_0(t)=\sigma\lambda_1(Tu_2)(t)+u_0(t),
\]
where $u_0\in P$ is defined by
$u_0(t)=M\int_0^{\infty}G(t,s)h(s){\rm\,d}s$. Notice
$r(\sigma\lambda_1T)=\lambda_1\sigma r(T)<1$. This implies
$I-\sigma\lambda_1 T$ is invertible  and its inverse equals
\[
(I-\sigma\lambda_1 T)^{-1}=I+\sigma\lambda_1 T
+\sigma^2\lambda_1^2 T^2+\dots+\sigma^n\lambda_1^n T^n+\dots.
\]
Now we have $(I-\sigma\lambda_1 T)^{-1}(P)\subset P$ and $u_2\leq
(I-\sigma\lambda_1 T)^{-1}u_0$. Therefore, $\mathcal{M}_1$ is
bounded. Choosing $R>\max\{r,\sup\{\|u\|: u\in\mathcal M_1\}\}$, we
have by Lemma \ref{lemma3}
\begin{equation}\label{index1-1}
i(A,B_{R}\cap P,P)=1.
\end{equation}
Now \eqref{index1-0} and \eqref{index1-1} imply
\begin{equation}
i(A,(B_{R}\setminus \overline{B}_r)\cap P,P)=i(A,B_{R}\cap
P,P)-i(A,B_r\cap P,P)=1.
\end{equation}
Thus  the operator $A$ has at least one fixed point on
$(B_{R}\setminus \overline{B}_r)\cap P$ and hence \eqref{problem}
has at least one positive solution. The proof is completed.
\end{proof}

\begin{theorem}\label{thm2}
Let {\rm (H1)-(H3)} hold. Suppose
\begin{equation}\label{superlinear-1}
\liminf_{u\to \infty}\frac{f(t,u)}{u}> \lambda_1
\end{equation}
 and
\begin{equation}\label{superlinear-2}
\limsup_{u\to 0^+}\frac{f(t,u)}{u}< \lambda_1
\end{equation}
uniformly for $t\in \mathbb{R}^+$. Then  \eqref{problem} has
at least one positive solution.
\end{theorem}

\begin{proof}
By \eqref{superlinear-1} and (H3), there exist
$\varepsilon > 0$ and $b> 0$ such that
\begin{equation}\label{fsuper}
f(t,u)\geq (\lambda_1 + \varepsilon)u - b, \quad \forall u\geq 0, \;
t\in \mathbb{R}^+.
\end{equation}
This implies
\begin{equation}\label{Ineq-2}
(Au)(t)\ge(\lambda_1+\varepsilon)\int_0^{\infty}
G(t,s)h(s)u(s){\rm\,d}s-b\int_0^{\infty}G(t,s)h(s){\rm\,d}s
\end{equation}
for all $u\in P$ and $t\in \mathbb{R}^+$.
We shall prove that the set
\begin{equation}\label{notequ-2}
\mathcal{M}_2:=\{u\in P: u=Au+ \mu\varphi, \mu\geq 0\}.
\end{equation}
is bounded, where $\varphi\in P$ is given by \eqref{Krein-Rutman-1}.
Indeed, if $u\in \mathcal{M}_2$, then we have $u\geq Au$
by definition and $u\in P_0$ by Lemma \ref{lemma4}.
This together with \eqref{Ineq-2}
leads to
\[
u(t)\ge (\lambda_1+\varepsilon)\int_0^{\infty}
G(t,s)h(s)u(s){\rm\,d}s-b\int_0^{\infty}G(t,s)h(s){\rm\,d}s, \quad
t\in\mathbb{R}^+.
\]
Multiply the above  by $\psi (t)$  and integrate over
$\mathbb{R}^+$ and use \eqref{Krein-Rutman-2} and
\eqref{integral} to obtain
\[
\int_0^{\infty} \psi(t)u(t){\rm\,d}t \geq(\lambda_1 +
\varepsilon)\lambda_1^{-1}\int_0^{\infty}
\psi(t)u(t){\rm\,d}t-b\lambda_1^{-1},
\]
so that $\int_0^{\infty} \psi(t)u(t){\rm\,d}t\leq b\varepsilon^{-1}$
for all $u\in\mathcal M_2$. Recalling  $u\in P_0$, we  obtain
 $\|u\|\leq (\varepsilon\omega)^{-1}b$ for all $u\in \mathcal M_2$,  and thus $\mathcal M_2$ is bounded, as required.
  Taking
$R>\sup\{\|u\|: u\in\mathcal M_2\}$, we have
\begin{equation}
 u-Au\neq \mu\varphi, \quad \forall u\in \partial B_R\cap P,\;
 \mu\geq 0.
\end{equation}
Now  Lemma \ref{lemma2} yields
\begin{equation}\label{index2-0}
i(A,B_R\cap P,P)=0.
\end{equation}
By \eqref{superlinear-2},  there exist $ r\in (0, R)$ and $\sigma\in
(0,\lambda_1)$ such that
\[
f(t,u) \leq (\lambda_1-\sigma)u, \quad
\forall 0 \leq u \leq r,\;  0\leq t<\infty,
\]
so that
\begin{equation}\label{Inequ-1}
(Au)(t)\leq (\lambda_1-\sigma)\int_0^\infty G(t,s)h(s)u(s){\rm\,d}s
\end{equation}
for all $u\in\overline B_r\cap P$ and $t\in \mathbb{R}^+$.
 We claim that
\begin{equation}\label{hom}
u\neq \mu Au, \quad \forall u\in \partial B_r\cap P,\quad
 0\leq \mu\leq 1.
\end{equation}
Suppose, to the contrary,  there exist $u_1 \in \partial B_r \cap P$
and $\mu_1 \in[0,1]$ such that $u_1 = \mu_1Au_1$. Then we  have
$u_1\in P_0$ by Lemma \ref{lemma4} and
\[
u_1(t)\leq (\lambda_1-\sigma)\int_0^\infty
G(t,s)h(s)u_1(s){\rm\,d}s, \quad t\in\mathbb{R}^+
\]
by \eqref{Inequ-1}. Multiply the above  by $\psi (t)$
and integrate over $\mathbb{R}^+$ and use \eqref{Krein-Rutman-2}
and \eqref{integral} to obtain
\[
\int_0^{\infty} \psi(t)u_1(t){\rm\,d}t
\leq \frac{\lambda_1-\sigma}{\lambda_1}
\int_0^{\infty} \psi(t)u_1(t){\rm\,d}t,
\]
so that $\int_0^{\infty} \psi(t)u_1(t){\rm\,d}t=0$.
Recalling $u_1\in P_0$, we obtain $u_1= 0$,
a contradiction with   $u_1\in \partial B_r\cap P$. As a result,
\eqref{hom} is true. Now  Lemma \ref{lemma3} yields
\begin{equation} \label{index2-1}
i(A,B_r\cap P,P)=1.
\end{equation}
Combining \eqref{index2-0} and \eqref{index2-1} gives
\begin{equation}
i(A,(B_R\setminus \overline{B}_r)\cap P,P)=i(A,B_R\cap
P,P)-i(A,B_r\cap P,P)=-1.
\end{equation}
Consequently the operator $A$ has at least one fixed point on
$(B_R\setminus \overline{B}_r)\cap P$, and hence \eqref{problem} has
at least one positive solution. The proof is completed.
\end{proof}

\begin{theorem}\label{thm3}
Let  {\rm (H1)-(H3)} hold.  Suppose  that $f(t,u)$
satisfies \eqref{sublinear-1} and \eqref{superlinear-1}.
Moreover, $f(t,u)$ is nondecreasing in $u$, and that
there exists $N> 0$ such that
\begin{equation}\label{nondecreasing}
f(t,N)<\frac{N}{\kappa},\quad\text{a.e. }
t\in\mathbb{R}^+,
\end{equation}
where $\kappa:=\int_0^{\infty} G(s,s)h(s){\rm\,d}s>0.$ Then
\eqref{problem} has at least two positive solutions.
\end{theorem}

\begin{proof}
The monotonicity of $f$ implies that for all $u\in \overline B_N\cap
P$ and $t\in \mathbb{R}^+$, we have
\begin{equation}
 (Au)(t)=\int_0^{\infty} G(t,s)h(s)f(s,u(s)){\rm\,d}s
< \int_0^{\infty} G(s,s)h(s)\frac{N}{\kappa}{\rm\,d}s= N,
\end{equation}
so that  $\|Au\|< \|u\|$ for all $u\in \partial B_N\cap P$. A
consequence of this is
\[
u\neq \mu Au,  \forall u\in \partial B_N\cap P, 0\leq \mu\leq 1.
\]
Now Lemma \ref{lemma3} implies
\begin{equation}\label{index3-1}
 i(A,B_N\cap P,P)=1.
\end{equation}
On the other hand, in view of \eqref{sublinear-1} and
\eqref{superlinear-1}, we may take
 $R>N$ and  $r\in (0,N)$ so
that \eqref{index1-0} and \eqref{index2-0} hold (see the proofs of
Theorems \ref{thm1} and \ref{thm2}). Combining  \eqref{index1-0},
\eqref{index2-0} and \eqref{index3-1}, we arrive at
\[
i(A,(B_R\backslash \overline{B}_{N})\cap P,P)=0-1=-1,\ \
i(A,(B_N\backslash \overline{B}_{r})\cap P,P)=1-0=1.
\]
Consequently the operator $A$ has at least two fixed points, one  on
$(B_R\backslash \overline{B}_{N})\cap P$ and the other on
$(B_N\backslash \overline{B}_{r})\cap P$.  Hence (1.1)  has at least
two positive solutions. The proof is completed.
\end{proof}

\section{Examples}
In this section, we provide four examples to illustrate applications
of Theorems \ref{thm1}--\ref{thm3}.
 Let us consider the  boundary value  problem
 \begin{equation}\label{example}
\begin{gathered}
 ((1+t^2)u'(t))'+t^{-1/2}e^{-t}f(t,u)=0,\quad  0<t<\infty,\\
 u(0)= u(\infty)=0,
\end{gathered}
\end{equation}
where $f\in C(\mathbb{R}^+\times \mathbb{R}^+,\mathbb{R}^+)$
satisfies (H3). Now we have  $p(t)=1+t^2$, $h(t)=t^{-1/2}e^{-t}$,
$a=c=1$, $b=d=0$,
\[
G(t,s)=\frac{2}{\pi}\begin{cases}
 (\frac{\pi}{2}- \arctan(t))\arctan(s), & 0\le s\le t<\infty,\\
 (\frac{\pi}{2}- \arctan(s))\arctan(t), & 0\le t\le s<\infty,
\end{cases}
\]
and
\[
\gamma (t)=\frac{2}{\pi}\min\left\{\frac{\pi}{2}-\arctan(t), \arctan(t) \right\},
 t\in\mathbb{R}^+.
\]
Since
\[
\int_0^{\infty}\frac{{\rm\,d}r}{p(r)}=\frac{\pi}{2}<\infty,
\kappa=\int_0^{\infty}G(s,s)h(s){\rm\,d}s<
\frac{\pi^{3/2}}{8}<\infty,
\]
 conditions (H1)-(H3) hold.
By elementary calculus, we have
\[
\arctan(s)\geq se^{-2s},\quad \frac{\pi}{2}-\arctan(s)\geq se^{-2s},
\quad s\in\mathbb{R}^+.
\]
The inequalities above, along with Gelfand's theorem, enable us to
derive the estimation $8\pi^{-3/2}< \lambda_1 <
\frac{1372\sqrt{7}}{15} \pi^{3/2}$, where $\lambda_1$ denotes the
first eigenvalue of the eigenvalue problem associated with
\eqref{example}.

\begin{example} \label{exa4.1} \rm
Let $f(t,u):=u^\alpha, t,u\in\mathbb{R}^+$, where $\alpha\in
(0,1)\cup (1,\infty)$.  If $\alpha\in (0,1)$, then
\eqref{sublinear-1} and \eqref{sublinear-2} are satisfied. If
$\alpha\in (1,\infty)$, then \eqref{superlinear-1} and
\eqref{superlinear-2} are satisfied. By Theorems \ref{thm1} and
\ref{thm2}, Equation \eqref{example} has at least one positive
solution.
\end{example}

\begin{example} \label{exa4.2}\rm
 Let \[f(t,u):=\begin{cases}
2\lambda_1u, & 0\leq u\leq 1,\\
\frac{\lambda_1u}{2}+\frac{3\lambda_1}{2},& u\geq 1.
\end{cases}
\]
Now \eqref{sublinear-1} and  \eqref{sublinear-2} are satisfied. By
Theorem \ref{thm1}, Equation \eqref{example} has at least one
 positive solution.
\end{example}

\begin{example} \label{exa4.3} \rm
Let
\[
f(t,u):=\begin{cases}\frac{\lambda_1u}{2}, &0\leq u\leq 1,\\
2\lambda_1u-\frac{3\lambda_1}{2},& u\geq 1.
\end{cases}
\]
Now \eqref{superlinear-1} and \eqref{superlinear-2} are satisfied.
By Theorem \ref{thm2}, Equation \eqref{example} has at least one
positive solution.
\end{example}

\begin{example} \label{exa4.4} \rm
Let $f(t,u):=\lambda (u^a+u^b)$, where $0<a<1<b$, $0<\lambda\leq 4
\pi^{-3/2}$. Now   \eqref{sublinear-1}, \eqref{superlinear-1} and
\eqref{nondecreasing} are satisfied. By Theorem \ref{thm3}, Equation
\eqref{example} has at least two positive solutions.
\end{example}


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