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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 74, pp. 1--25.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/74\hfil Multiple positive solutions]
{Multiple positive solutions for singular boundary-value
problems with derivative dependence on finite and infinite
intervals}
\author[B. Yan\hfil EJDE-2006/74\hfilneg]
{Baoqiang Yan}

\address{Baoqiang Yan \newline
Department of Mathematics, Shandong Normal University, Ji-nan,
250014 China\newline
and School of Mathematics, Trinity College, Dublin University, Dublin
2, Ireland}
\email{bqyan819@beelink.com}

\date{}
\thanks{Submitted March 16, 2006. Published July 10, 2006.}
\thanks{Supported by grants 10571111 from the  National Natural
  Science and Y2005A07 from the \hfill\break\indent
  natural science of Shandong Province}
\subjclass[2000]{34B16, 34B18, 34B40} 
\keywords{Krasnoselskii's theorem; singular boundary value problems; \hfill\break\indent
fixed point theorem of cone expansion and compression}

\begin{abstract}
 In this paper, Krasnoselskii's  theorem and the fixed point
 theorem of cone expansion and compression are improved.
 Using the results obtained, we establish the existence of
 multiple positive solutions  for the singular second-order boundary-value
 problems with derivative dependance on finite and infinite intervals.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}


In \cite{a1}, by an alternative method to Leray-Schauder and
sequential technique, Agarwal and O'Regan considered the singular
boundary-value problem
\begin{equation}
\begin{gathered}
\frac{1}{p}(py')'+\Phi(t)f(t,y,py')=0, \quad  t\in(0,1)\\
 \alpha y(0)-\beta\lim_{t\to 0^+}p(t)y'(t)=0,\quad  y(1)=0
\end{gathered} \label{e1.1}
\end{equation}
and obtained the existence of one solution
to equation \eqref{e1.1} when $\alpha=0$ or $\beta=0$.

In \cite{p1},  by a generalization of the Kneser's property
(continuum) of the cross-sections of the solutions funnel,
Palamides and  Galanis considered the following problems
\begin{equation}
\begin{gathered}
\frac{1}{p}(py')'+\Phi(t)f(t,y,py')=0, \quad  t\in(0,+\infty)\\
 y(0)=0,\quad \lim_{t\to +\infty}p(t)y'(t)=0
\end{gathered} \label{e1.2}
\end{equation}
and also obtained the existence of one
positive and monotone unbounded solution.

There are some other results on the existence of at least one
solution for equation \eqref{e1.1}, \eqref{e1.2}, and we refer the reader also
to \cite{a2,a3,b1,b2,b3,b4,c1,g1,o1,o2,o3,o4}.
Moreover, under the condition that $p\equiv 1$,
$\beta=0$ and $f$ has no singularity at $x=0$ and $px'=0$, in
\cite{h1}, using pairs of lower and upper solutions, Henderson and
Thompson considered the existence of three solutions for
equation \eqref{e1.1} and in \cite{y1}, by reducing the equation \eqref{e1.1} to a
quasi-linear one, I.Yermachenko and F.Sadyrbaev obtained the
existence of multiple solutions for equation \eqref{e1.1} also.

Up to now, there are fewer results on the existence of multiple
positive solutions to equation \eqref{e1.1}, \eqref{e1.2} if $f(t,x,px')$ is
singular at $x$ and is related to $px'$. Motivated by this, in
this paper, we discuss the existence of multiple positive
solutions to equation \eqref{e1.1}, \eqref{e1.2} when $f(t,x,px')$ is singular
at $x=0$.


There are  three  sections in our paper. In section 2, in order
to overcome the difficulty from $px'$, we improve the
Krasnoselskii's theorem and fixed point theorem of cone expansion
and compression on unbounded set in a Banach space with a special
norm. In section 3, we establish special cones, and using obtained
theorems, present the existence of multiple positive solutions to
equation \eqref{e1.1}. In section 4, we consider the existence of
multiple positive solutions to equation \eqref{e1.2}.

\section{The improvement of the Krasnoselskii's  theorem and
 fixed point theorem of cone expansion and compression}

In this section, we improve the Krasnoselskii's  theorem and fixed
point theorem of cone expansion and compression in a Banach space
with a special norm.

In \cite{g2}, Granas and Dugudji presented the theory of fixed
point index on unbounded open sets which has same basic properties
as those in the theory of fixed point index on bounded open
sets \cite{g4}. The degree theory on bounded open sets and
unbounded open sets can be found in \cite{d1,g2,g3,k1,l1,z1}.

According to the the theory of fixed point index on unbounded open
sets in Chapter 4 of \cite{g2}, it is easy to obtain following result.
Let $E$ be a real Banach space containing a cone $P$.

\begin{lemma} \label{lem2.1}
 Assume $\Omega\subseteq E$,
$\theta\in\Omega$, $\Omega\cap P$ is a relatively open set in $P$.
Let $A:P\cap\overline{\Omega}\to P$ be continuous with relatively
compact $A(P\cap\overline{\Omega})$. Suppose that
\begin{equation}
Ax\neq \mu x,\quad  \forall x\in P\cap\partial\Omega,\;
 \mu\geq 1.\label{e2.1}
\end{equation}
Then $i(A,P\cap\Omega,P)=1$.
\end{lemma}

\begin{proof}  Let $H(t,x)=tAx$, $t\in[0,1]$ and $x\in
P\cap\partial\Omega$. Then
$H:[0,1]\times(P\cap\overline{\Omega})\to P$ is continuous, and
the continuity of $H(t,x)$ in $t$ is uniform with respect to $x\in
P\cap\overline{\Omega}$. Moreover, $H(t,P\cap\overline{\Omega})$
is relatively compact for every $t\in[0,1]$. Evidently,
$H(t,x)\neq x$ for $x\in P\cap\partial\Omega$ and $0\leq t\leq 1$.
Hence, by the homotopy invariance and normality of fixed point
index, we have
$$
i(A,P\cap\Omega,P)=i(\theta,P\cap\Omega,P)=1.
$$
The proof is complete.
\end{proof}

Now we consider a real Banach space in a special case.
Assume that $E$ is a linear space and it satisfies three
conditions:
\begin{enumerate}
\item There is a norm $x\to \|x\|_1$ on $x\in E$ and under
$\|\cdot\|_1$, $E$ is a normed linear space (not complete)

\item There is another semi-norm $\|\cdot\|_2$

\item Under $\|x\|=\max\{\|x\|_1,\|x\|_2\}$, $E$ is a Banach
space.
\end{enumerate}
For example, for $x\in C^1([0,1],R)$, under
$\|x\|_1=\max_{t\in[0,1]}|x(t)|$, $C^1([0,1],R)$ is an
incomplete normed linear space. Let
$\|x\|_2=\max_{t\in[0,1]}|x'(t)|$. Obviously, $\|\cdot\|_2$ is
a semi-norm of $C^1([0,1],R)$. If we define
$\|x\|=\max\{\|x\|_1,\|x\|_2\}$, $C^1([0,1],R)$ is a Banach space.


Assume that $P$ is a cone of $E$ and $\Omega\subset E$ is a open
set with $\sup_{x\in\overline{\Omega}}\|x\|_1<+\infty$. Since
$\sup_{x\in\overline{\Omega}}\|x\|_1\leq
\sup_{x\in\overline{\Omega}}\|x\|$, it is possible that $\Omega$
is unbounded in $E$. We have the following lemma(the ideas coming
from \cite{g4}).

 \begin{lemma} \label{lem2.2}
With $E$ and $P$ as above,
 assume that $\Omega\subseteq E$ is an open set with
$\sup_{x\in\overline{\Omega}}\|x\|_1<+\infty$. Let $A:P\cap
\overline{\Omega}\to P$ be continuous with relatively compact
$A(P\cap\overline{\Omega})$ and $B:P\cap\partial\Omega\to P$ be
continuous with relatively compact $B(P\cap\partial {\Omega})$.
Suppose that
\begin{itemize}
\item[(a)] $ \inf_{x\in P\cap\partial\Omega}\|Bx\|_1>0$;

\item[(b)] $x-Ax\neq tBx$, for all $x\in P\cap\partial\Omega$,
$t\geq 0$.
\end{itemize}
Then, we have
\begin{equation}
 i(A,P\cap \Omega,P)=0.\label{e2.2}
\end{equation}
\end{lemma}

\begin{proof}  Suppose that the $E_1$ is a Banach space completion  of $E$
under norm $\|x\|_1$. By the extension
theorem of Dugundji \cite{d2}, we can extend $B$ to a continuous
operator from $P\cap\overline{\Omega}$ into $P$ such that
\begin{equation}
B(P\cap\overline{\Omega})\subseteq \overline{{\rm co}}
B(P\cap\partial(\Omega))
\subseteq (\overline{{\rm co}}B(P\cap \partial\Omega))_1,\label{e2.3}
\end{equation}
where $(\overline{{\rm co}}B(P\cap \partial\Omega))_1$ is the closure of
$B(P\cap \partial\Omega)$ under the norm $\|\cdot\|_1$ and the
followings are similar. Let $F=B(P\cap\partial\Omega)$, then
$(\overline{{\rm co}}B(P\cap\partial\Omega))_1=(\overline{{\rm co}}F)_1
=(\overline{M})_1$,
where
$$
M=\{y=\sum_{i=1}^n\lambda_iy_i : y_i\in F,\lambda_i\geq
0,\sum_{i=1}^n\lambda_i=1; n=1,2,\dots\}.
$$
We first prove
\begin{equation}
\inf_{y\in(\overline{M})_1}\|y\|_1>0.\label{e2.4}
\end{equation}
Denote by $E_0$ the subspace of $E$ spanned by $F$ under norm
$\|\cdot\|_1$. Since $B(P\cap\partial\Omega)$ is relatively
compact in $E$ under norm $\|\cdot\|$, we know that
$B(P\cap\partial\Omega)$ is relatively compact in $E_1$ under norm
$\|\cdot\|_1$. Therefore, $E_0$ is separable. Evidently,
$P_0=P\cap E_0$ is a cone of $E_0$ and $F\subseteq P_0$. By
property of the cone \cite[Theorem 1.4.1]{g4}, there exists
$f_0\in E_0^*$ such that $f_0(y)>0$ for any $y\in P_0$ with
$y\neq \theta$. We claim that
\begin{equation}
\inf_{y\in F}f_0(y)=\sigma>0.\label{e2.5}
\end{equation}
In fact, if $\sigma=0$, then there exists $\{y_k\}\subseteq F$
such that $f_0(y_k)\to 0$. By the relative compactness of $F$ in
$E$, there is a subsequence $\{y_{k_i}\}$ of $\{y_k\}$ such that
$y_{k_i}\to y_0\in P$ and $y_0\in E_0$. Then $y_0\in P_0$, and so
$f_0(y_{k_i})\to f_0(y_0)=0$. Hence, $y_0=\theta$ and
$\|y_{k_i}\|_1\to 0$, which contradicts hypothesis (a). Thus,
\eqref{e2.5} holds.

For any  $ y=\sum_{i=1}^{n}\lambda_iy_i\in M$, where $y_i\in
F$, $\lambda_i\geq 0$ and $\sum_{i=1}^n\lambda_i=1$, we have
$$
f_0(y)=\sum_{i=1}^n\lambda_if_0(y_i)\geq
\sum_{i=1}^n\lambda_i\sigma=\sigma,
$$
and therefore
\begin{equation}
f_0(y)\geq\sigma, \forall y\in(\overline{M})_1.\label{e2.6}
\end{equation}
Since $(\overline{M})_1=(\overline{{\rm co}}F)_1$ is compact, there
exists a $z_0\in(\overline{M})_1$ such that
\begin{equation}
\inf_{y\in(\overline{M})_1}\|y\|_1=\|z_0\|_1.\label{e2.7}
\end{equation}
By \eqref{e2.6}, $f_0(z_0)\geq\sigma$, and this implies that
$z_0\neq \theta$. It follows therefore from \eqref{e2.7} that \eqref{e2.4}
holds. By \eqref{e2.3} and \eqref{e2.4}, we get
\begin{equation}
\inf_{x\in P\cap{\overline{\Omega}}}\|Bx\|_1=\sigma>0.\label{e2.8}
\end{equation}
Now, it is easy to show that \eqref{e2.2} holds. In fact, if
$i(A,P\cap\Omega,P)\neq 0$, then by the hypothesis (b) and the
homotopy invariance property of fixed point index, we have
$$
i(A+tB,P\cap\Omega,P)=i(A,P\cap\Omega,P)\neq 0,\forall t>0.
$$
In particular, choosing $t_0>\frac{a+c}{\sigma}$, where
$a=\sup_{x\in\overline{\Omega}}\|x\|_1$ and $c=\sup_{x\in
P\cap\overline{\Omega}}\|Ax\|_1$, we have
$$
i(A+t_0B, P\cap\Omega,P)\neq 0,
$$
and so, by the solution property of fixed point index, there
exists an $x_0\in P\cap\Omega$ such that $Ax_0+t_0Bx_0=x_0$. Hence
$$
t_0=\frac{\|x_0-Ax_0\|_1}{\|Bx_0\|_1}\leq\frac{a+c}{\sigma},
$$
which is a contradiction. The proof is complete.
\end{proof}

\begin{corollary} \label{coro2.1}
Assume that $\Omega$ is an open set with
$\sup_{x\in\overline{\Omega}}\|x\|_1<+\infty$. Let
$A:P\cap\overline{\Omega}\to P$ be continuous with relatively
compact $A(P\cap\overline{\Omega})$. If there exists $u_0>\theta$
such that
\begin{equation}
x-Ax\neq t u_0, \forall x\in P\cap\partial\Omega, t\geq 0,\label{e2.9}
\end{equation}
then \eqref{e2.2} holds.
\end{corollary}

\begin{proof}  Since $\|\cdot\|_1$ is a norm, $u_0>\theta$ implies that
 $\|u_0\|_1>0$. Hence, the corollary follows directly from Lemma
 \ref{lem2.2} by putting
 $Bx=u_0$ for any $x\in P\cap\partial\Omega$.
\end{proof}


\begin{corollary} \label{coro2.2}
Assume that $\Omega$ is an open set with
$\sup_{x\in\overline{\Omega}}\|x\|_1<+\infty$. Let
$A:P\cap\overline{\Omega}\to P$ be continuous with relatively
compact $A(P\cap\overline{\Omega})$. If
\begin{equation}
Ax\not\leq x, \quad \forall x\in P\cap\partial\Omega
,\label{e2.10}
\end{equation}
then \eqref{e2.2} holds.
\end{corollary}

\begin{proof}  Choose an $u_0>\theta$. Then,
$$
x-Ax\neq t u_0, \forall x\in P\cap\partial\Omega, \ t\geq 0.
$$
By Corollary \ref{coro2.1}, \eqref{e2.2} holds.
\end{proof}

\begin{lemma} \label{lem2.3}
Assume that $\Omega$ is an open set with
$\sup_{x\in\overline{\Omega}}\|x\|_1<+\infty$. Let
$A:P\cap\overline{\Omega}\to P$ be continuous with relatively
 compact $A(P\cap\overline{\Omega})$. Suppose that
\begin{itemize}
\item[(i)] $\inf_{x\in P\cap\partial\Omega}\|Ax\|_1>0$; and
\item[(ii)] $Ax\neq \mu x,\forall x\in P\cap\partial\Omega,0<\mu<1$.
\end{itemize}
 Then \eqref{e2.2} holds.
\end{lemma}

 \begin{proof}  Taking $B=A$ in Lemma \ref{lem2.2}, we see that condition (a) of
 Lemma \ref{lem2.2} is the same as condition $(i)$ of Lemma \ref{lem2.3}. Also,
 condition (b) of Lemma \ref{lem2.2} is true. In fact, if there exist
 $x_0\in P\cap\partial\Omega$ and $t_0\geq 0$ such that
 $x_0-Ax_0=t_0Ax_0$, then $Ax_0=\mu x_0$, where
 $\mu_0=(1+t_0)^{-1}$. Evidently $0<\mu_0\leq 1$, which
 contradicts the condition $(ii)$. Thus, \eqref{e2.2} follows
from Lemma \ref{lem2.2}.
 \end{proof}

 \begin{lemma} \label{lem2.4}
Assume that $\Omega$ is an open set with
$\sup_{x\in\overline{\Omega}}\|x\|_1<+\infty$. Let
$A:P\cap\overline{\Omega}\to P$ be continuous with relatively
compact  $A(P\cap\overline{\Omega})$. Suppose that
\begin{itemize}
\item[(i')]
 $Ax\neq \mu x, \forall x\in P\cap\partial\Omega, 0\leq\mu\leq 1$,
 and
\item[(ii')] the set $\{\|Ax\|_1^{-1}Ax|x\in P\cap\partial\Omega\}$ is
 relatively compact.
\end{itemize}
 Then \eqref{e2.2} holds.
\end{lemma}

\begin{proof}  Let $A_1x=\alpha(\|Ax\|_1)^{-1}Ax$  for
$x\in P\cap\partial\Omega$, where  $\alpha=\sup_{x\in
P\cap\partial\Omega}\|Ax\|_1>0$. Then, by hypotheses,
$A_1:P\cap\partial\Omega\to P$ is continuous with relatively
compact $A_1(P\cap\partial{\Omega})$. By the extension theorem,
$A_1$ can be extended to a continuous operator from
$P\cap\overline{\Omega}$ into $P$ with relatively compact
$A_1(P\cap{\Omega})$. We now prove that $A_1$ satisfies the
condition (i) and (ii) of Lemma \ref{lem2.3}. In fact, first we have
$$
\inf_{x\in P\cap\partial\Omega}\|A_1x\|_1=\sigma>0.
$$
Secondly, if there exists $x_0\in P\cap\partial\Omega$ and
$0<\mu_0\leq 1$ such that $A_1x_0=\mu_0 x_0$, then
$Ax_0=\lambda_0x_0$, where $\lambda_0=\mu_0\alpha^{-1}\|Ax_0\|_1$.
Evidently, $0<\lambda_0\leq\mu_0\leq 1$, which contradicts
hypothesis (i'). Hence, by Lemma \ref{lem2.3}, we have
\begin{equation}
i(A_1,P\cap\Omega,P)=0.\label{e2.11}
\end{equation}
Now, we prove
\begin{equation}
(1-t)Ax+tA_1x\neq x, \forall x\in
P\cap\partial\Omega, 0\leq t\leq 1.\label{e2.12}
\end{equation}
 If there is an $x_1\in P\cap\partial\Omega$ and a
$0\leq t_1\leq  1$ such that $(1-t_1)Ax_1+t_1A_1x_1=x_1$,
then $Ax_1=\mu_1x_1$,
 where $\mu_1=[1+t_1(\alpha/\|Ax_1\|_1-1)]^{-1}$, $0\leq \mu_1\leq 1$,
in contradiction with hypothesis (i').
Hence, by \eqref{e2.11}, \eqref{e2.12} and the homotopy invariance
of fixed point index, we get
 $$
i(A,P\cap \Omega,P)=i(A_1,P\cap\Omega,P)=0.
$$
 The proof is complete.
\end{proof}

\begin{theorem} \label{thm2.5}
Let $\Omega_1$ and $\Omega_2$ be
two open in $E$ such that $\theta\in\Omega_1$ and
$\overline{\Omega}_1\subseteq\Omega_2$ with
$\sup_{t\in\overline{\Omega}_2}\|x\|_1<+\infty$. Let
$A:P\cap(\overline{\Omega}_2-\Omega_1)\to P$ be continuous with
relatively compact $A(P\cap(\overline{\Omega}_2-\Omega_1))$.
Suppose that one of the two conditions
\begin{itemize}
\item[(H1)]  $Ax\not\geq x$, $\forall x\in P\cap\partial\Omega_1$ and
$Ax\not\leq x$, $\forall x\in P\cap\partial\Omega_2$,

\item[(H2)] $Ax\not\leq x$, $\forall x\in P\cap\partial\Omega_1$ and
$Ax\not\geq x$, $\forall x\in P\cap\partial\Omega_2$
\end{itemize}
is satisfied. Then $A$ has at least one fixed point in
$P\cap(\Omega_2-\overline{\Omega}_1)$.
\end{theorem}

\begin{proof}  By the extension theorem (Dugundji \cite{d2}), $A$ has a
completely continuous extension (also noted by $A$) from
$P\cap\overline{\Omega}_2$ from $P\cap\overline{\Omega}_2$ to $P$.
First we assume that (H1) is satisfied, i.e., it is the case of
cone expansion. It is easy to see that
\begin{equation}
Ax\neq \mu x,\quad \forall x\in
P\cap\partial\Omega_1,\; \mu\geq1,\label{e2.13}
\end{equation}
since, otherwise, there exists $x_0\in P\cap\partial\Omega_1$ and
$\mu_0\geq 1$ such that $Ax_0=\mu_0x_0\geq x_0$, in contradiction
with (H1). Now, from \eqref{e2.1} and Lemma \ref{lem2.1}, we obtain
\begin{equation}
i(A,P\cap \Omega_1,P)=1.\label{e2.14}
\end{equation}
On the other hand, by Corollary \ref{coro2.2}, we have
\begin{equation}
i(A,P\cap \Omega_2,P)=0.\label{e2.15}
\end{equation}
It follows therefore from \eqref{e2.14} and
\eqref{e2.15} and additivity property of fixed point index that
\begin{equation}
i(A,P\cap(\Omega_2-\overline{\Omega}_1),P)
=i(A,P\cap\Omega_2,P)-i(A,P\cap\Omega_1,P)=-1\neq 0.\label{e2.16}
\end{equation}
Hence, by the solution property of fixed point index, $A$ has at
least one fixed point in $P\cap(\Omega_2-\overline{\Omega}_1)$.

Similarly, when (H2) is satisfied, instead of \eqref{e2.14}, \eqref{e2.15},
we have $i(A,P\cap\Omega_2,P)=1$, and
$i(A,P\cap(\Omega_2-\overline{\Omega}_1),P)=1$. As a result we
also can assert that $A$ has at least one fixed point in
$P\cap(\Omega_2-\overline{\Omega}_1)$. The proof is complete.
\end{proof}

We remark that this theorem improves  \cite[theorem 2.3.3]{g4}
because the condition that $\Omega_1$ and $\Omega_2$ are bounded
is not necessary.

\begin{theorem} \label{thm2.6}
Let $\Omega_1=\{x\in E|\|x\|_1<r\}$ and
$\Omega_2=\{x\in E|\|x\|_1<R\}$ be two open in $E$ with $r<R$. Let
$A:P\cap(\overline{\Omega}_2-\Omega_1)\to P$ be continuous with
relatively compact $A(P\cap(\overline{\Omega}_2-\Omega_1))$.
Suppose that one of the two conditions
\begin{itemize}
\item[(H3)] $\|Ax\|\leq \|x\|$, for all $x\in P\cap\partial\Omega_1$
and $\|Ax\|_1\geq\|x\|_1$, for all $x\in P\cap\partial\Omega_2$,

\item[(H4)] $\|Ax\|_1\geq\|x\|_1$, for all
$x\in P\cap\partial\Omega_1$ and $\|Ax\|\leq\|x\|$, for all
$x\in P\cap\partial\Omega_2$
\end{itemize}
is satisfied. Then $A$ has at least one fixed point in
$P\cap(\Omega_2-\overline{\Omega}_1)$.
\end{theorem}

\begin{proof}  We only need to prove  this theorem under condition
(H3), since the proof is similar when (H4) is satisfied. By the
extension theorem, $A$ can be extended to a continuous operator
from $P\cap\overline{\Omega}_2$ into $P$ with relatively compact
$A(P\cap\overline{\Omega}_2)$. We may assume that $A$ has no fixed
points on $P\cap\partial\Omega_1$ and $P\cap\partial\Omega_2$. It
is easy to see that \eqref{e2.13} holds, since otherwise, there
exist $x_0\in P\cap\partial\Omega_1$ and $\mu_0>1$ such that
$Ax_0=\mu_0x_0$ and hence $\|Ax_0\|=\mu_0\|x_0\|>\|x_0\|$, in
contradiction with (H3). Thus, by \eqref{e2.13}, Lemma
\ref{lem2.1}, \eqref{e2.14} holds.

On the other hand, it is also easy to verify
\begin{equation}
Ax\neq \mu x,\forall x\in P\cap\partial\Omega_2,0<\mu<1.\label{e2.17}
\end{equation}
In fact, if there are $x_1 \in P\cap\partial\Omega_2$ and
$0<\mu_1<1$ such that $Ax_1=\mu_1x_1$, then
$$
\|Ax_1\|_1=\mu_1\|x_1\|_1<\|x_1\|_1,
$$
in contradiction with (H3). In addition, by (H3) we have
\begin{equation}
\inf_{x\in P\cap\partial\Omega_2}\|Ax\|_1\geq
\inf_{x\in\partial\Omega_2}\|x\|_1>0.\label{e2.18}
\end{equation}
It follows from \eqref{e2.17}, \eqref{e2.18} and Lemma \ref{lem2.3}
that \eqref{e2.15} holds.
As before, \eqref{e2.14} and \eqref{e2.15} imply \eqref{e2.16},
and therefore $A$ has at least one fixed point in
$P\cap(\Omega_2-\overline{\Omega}_1)$.
\end{proof}

We remark that this theorem improves the the Krasnoselskii's
theorem in \cite{g4} because the condition that $\Omega_1$ and
$\Omega_2$ are bounded is not necessary.

\section{The existence of multiple positive solutions to equation \eqref{e1.1}}

In this section, we consider \eqref{e1.1} and suppose that $f\in
C([0,1]\times R^+_0\times R,R^+)$, $p\in C([0,1],R)\cap
C((0,1),R^+_0)\cap C^1((0,1),R)$ with
$\int_0^1\frac{1}{p(r)}dr<+\infty$, $\Phi\in C((0,1),R^+_0)$
and $\alpha\geq 0$, $\beta\geq 0$ (not equal to $0$ at the same
time); here $R^+=[0,+\infty)$, $R^+_0=(0,+\infty)$,
$R=(-\infty,+\infty)$.
Let
\begin{gather*}
\rho^2=\beta+\alpha\int_0^1\frac{1}{p(r)}dr \quad(\rho>0),\\
u_1(t)=\frac{1}{\rho}\int_t^1\frac{1}{p(r)}dr,\quad
v_1(t)=\frac{1}{\rho}(\beta+\alpha\int_0^t\frac{1}{p(r)}dr),\\
G_1(t,s)=\begin{cases} u_1(t)v_1(s)p(s),& 0\leq s\leq t\leq 1\\
v_1(t)u_1(s)p(s),& 0\leq t\leq s\leq 1.
\end{cases}
\end{gather*}
Assume that $C^1_p[0,1]=\{x:[0,1]\to R|$ $x(t)$ is continuous on
$[0,1]$ and $p(t){x'}(t)$ is continuous on $[0,1]$ also with
$\max_{t\in[0,1]}p(t)|{x'}(t)|<+\infty\}$ (see \cite{o3}).
For $x\in C^1_p$, let $\|x\|_1=\max_{t\in[0,1]}|x(t)|$,
$\|x\|_2=\max_{t\in[0,1]}p(t)|x'(t)|$ and
$\|x\|=\max\{\|x\|_1,\|x\|_2\}$. It is easy to see that $C^1_p$
satisfies the conditions (1), (2) and (3) of the Banach
space $E$ in section 2.


Obviously, $x(t)\in C^1_p$ is a solution to equation \eqref{e1.1} if and
only if $x(t)$ is a solution of the following integral equation
$$
x(t)=\int_0^1G_1(t,s)\Phi(s)f(s,x(s),p(s)x'(s))ds,\quad t\in[0,1].
$$
Let $ P=\{x\in C^1_p|x(t)\geq\gamma_1(t)\|x\|_1\}$, where
$\gamma_1(t)= u_1(t)v_1(t)\frac{1}{u_1(0)v_1(1)}$ for all
$t\in[0,1]$.

\begin{lemma} \label{lem3.1}
Assume that $l\in L^1[0,1]$ with $l(t)>0$ for
all $t\in(0,1)$ and  $q(t)=\int_0^1G_1(t,s)l(s)ds$,
$t\in[0,1]$. Then
$$
q(t)\geq\gamma_1(t)\max_{s\in[0,1]}q(s).
$$
\end{lemma}

\begin{proof} Suppose $q(t_0)=\max_{s\in[0,1]}q(s)$. Then
\begin{align*}
\frac{G_1(t,s)}{G_1(t_0,s)}
&=\begin{cases}
\frac{v_1(t)u_1(s)p(s)}{u_1(t_0)v_1(s)p(s)},&0\leq
t\leq s\leq t_0\leq 1\\
\frac{u_1(t)v_1(s)p(s)}{v_1(t_0)u_1(s)p(s)},&0\leq
t_0\leq s\leq t\leq  1\\
\frac{v_1(t)u_1(s)p(s)}{v_1(t_0)u_1(s)p(s)},&0\leq
t,t_0\leq s\leq 1\\
\frac{u_1(t)v_1(s)p(s)}{u_1(t_0)v_1(s)p(s)},&0\leq
s\leq t,t_0\leq 1\\
\end{cases}\\
&=\begin{cases}
u_1(t)v_1(t)\frac{u_1(s)}{u_1(t_0)}\frac{1}{v_1(s)u_1(t)},&0\leq
t\leq s\leq t_0\leq 1\\
u_1(t)v_1(t)\frac{v_1(s)}{v_1(t_0)}\frac{1}{u_1(s)v_1(t)},&0\leq
t_0\leq s\leq t\leq  1\\
u_1(t)v_1(t)\frac{1}{u_1(t)v_1(t_0)},&0\leq t,t_0\leq s\leq
1\\
 u_1(t)v_1(t)\frac{1}{u_1(t_0)v_1(t)},&0\leq
s\leq t,t_0\leq 1\\
\end{cases} \\
& \geq \begin{cases}
 u_1(t)v_1(t)\frac{1}{v_1(1)u_1(0)},&0\leq
t\leq s\leq t_0\leq 1\\
 u_1(t)v_1(t)\frac{1}{u_1(0)v_1(1)},&0\leq
t_0\leq s\leq t\leq  1\\
u_1(t)v_1(t)\frac{1}{u_1(0)v_1(1)},&0\leq t,t_0\leq s\leq
1\\
 u_1(t)v_1(t)\frac{1}{u_1(0)v_1(1)},&0\leq
s\leq t,t_0\leq 1\\
\end{cases}\\
&=\gamma_1(t).
\end{align*}
As a consequence,
\begin{align*}
q(t)&=\int_0^1G_1(t,s)l(s)ds
     =\int_0^1\frac{G_1(t,s)}{G_1(t_0,s)}G_1(t_0,s)l(s)ds \\
&\geq\gamma_1(t)\int_0^1G_1(t_0,s)l(s)ds=\gamma_1(t)\max_{s\in[0,1]}q(s).
\end{align*}
The proof is complete.
\end{proof}

Now we will list some conditions  for convenience:
\begin{itemize}
\item[(H1)] There exists a $k\in C([0,1],R^+_0)$,
 a $g\in C(R^+_0,R^+_0)$ and a decreasing continuous function $h\in
C(R^+_0, R^+_0)$ such that
$$
f(t,x,z)\leq k(t)g(x),\quad \forall x\in R^+_0, z\in
R, t\in[0,1],
$$
where $\frac{g(x)}{h(x)}$ is an increasing function and
$\int_0^{1}p(s)\Phi(s)k(s)h(c{\gamma}_1(s))ds<+\infty$ for each
$c>0$;

\item[(H2)]
$$
\sup_{c\in R^+_0}\frac{ch(c)}{u_1(0)v_1(1)\int_0^{1}p(s)
\Phi(s)k(s)h(c{\gamma}_1(s))ds g(c)}>1;
$$

\item[(H3)] There exists a $k_1\in
C([0,1],R^+_0)$ and a $g_1\in C(R^+_0,R^+_0)$ with
$f(t,x,z)\geq k_1(t)g_1(x)$, for all
$(t,x,z)\in[0,1]\times R^+_0\times (-\infty,+\infty)$ such that
$$
\lim_{x\to+\infty}\frac{g_1(x)}{x}=+\infty,
$$
where $ \int_0^{1}p(s)\Phi(s)k_1(s)ds<+\infty$;

\item[(H4)] For any  $c>0$,  there exists a $\psi_c\in C([0,1],R^+_0)$
such that $f(t,x,z)\geq \psi_{c}(t)$ for all $(t,x,z)\in
[0,1]\times (0,c]\times (-\infty,+\infty)$ with $
\int_0^{1}p(s)\Phi(s)\psi_c(s)ds<+\infty$.

\end{itemize}
For given $n\in\{1,2,\dots\}$, let
$f_n(t,x,z)=f(t,\max\{\frac{1}{n},x\},z)$
and for $x\in P$, define
\begin{equation}
(A_nx)(t)=\int_0^1G_1(t,s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds,\quad
 n\in\{1,2,\dots\},\; t\in[0,1].\label{e3.1}
\end{equation}

\begin{lemma} \label{lem3.2}
Assume  the condition (H1) holds.
Then, for every $n\in\{1,2,\dots\}$, $A_n:P\to P$ is continuous
and for any $r>0$ and $ B_r=\{x\in C^1_p|\|x\|_1\leq r\}$,
$A_n(P\cap B_r)$ is relatively compact.
\end{lemma}

\begin{proof}
First, for a given $n\in\{1,2,\dots\}$, we show that
$A_nP\subseteq P$. For any $x\in P$, we have
\begin{align*}
|(A_nx)(t)|&=\big|\int_0^1G_1(t,s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\big|\\
&\leq\int_0^1G_1(t,s)\Phi(s)f(s,\max\{\frac{1}{n},x(s)\},p(s){x'}(s))ds\\
&\leq\int_0^1G_1(t,s)\Phi(s)k(s)g(\max\{\frac{1}{n},x(s)\})ds\\
&\leq\int_0^1G_1(t,s)\Phi(s)k(s)h(\max\{\frac{1}{n},x(s)\})\frac{g(\max\{\frac{1}{n},x(s)\})}{h(\max\{\frac{1}{n},x(s)\})}ds\\
&\leq\int_0^1G_1(t,s)\Phi(s)k(s)ds
h(\frac{1}{n})\frac{g(\max\{\frac{1}{n},\|x\|_1\})}{h(\max\{\frac{1}{n},\|x\|_1\})}\\
&<+\infty
\end{align*}
and
\begin{align*}
|p(t)(A_nx)'(t)|
&=|-\frac{1}{\rho}\int_0^tv_1(s)p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\\
&\quad +\frac{\alpha}{\rho}\int_t^1u_1(s)p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds|\\
&\leq\frac{K}{\rho}\int_0^1p(s)\Phi(s)f(s,\max\{\frac{1}{n},x(s)\},p(s){x'}(s))ds\\
&\leq\frac{K}{\rho}\int_0^1p(s)\Phi(s)k(s)g(\max\{\frac{1}{n},x(s)\})ds\\
&\leq\frac{K}{\rho}\int_0^1p(s)\Phi(s)k(s)h(\max\{\frac{1}{n},x(s)\})\frac{g(\max\{\frac{1}{n},x(s)\})}{h(\max\{\frac{1}{n},x(s)\})}ds\\
&\leq\frac{K}{\rho}\int_0^1p(s)\Phi(s)k(s)ds
h(\frac{1}{n})\frac{g(\max\{\frac{1}{n},\|x\|_1\})}{h(\max\{\frac{1}{n},
\|x\|_1\})}\\ &<+\infty,
\end{align*}
where $K=\max\{\alpha u_1(0), v_1(1)\}$ and the following is
same as before. Then, $A_n$ is well defined. Moreover, from Lemma \ref{lem3.1},
for any $x\in P$, we have
$$
(A_nx)(t)\geq \gamma_1(t)\max_{s\in[0,1]}|(A_nx)(s)|
=\gamma_1(t)\|A_nx\|_1, \quad \forall t\in [0,1].
$$
Consequently, $A_nP\subseteq P$.

Second, we show $A_n:P\to P$ is continuous. Assume that
$\lim_{m\to+\infty}x_m=x_0$, which means there exists an
$M>1/n$ such that $\|x_m\|\leq M$, for all
$m\in\{0,1,2,\dots\}$. Then
\begin{equation}
\begin{aligned}
 f_n(t,x_m(t),p(t){x'}_m(t))
&\leq  k(t)g(\max\{\frac{1}{n},x_m(t)\})\\
&=k(t)h(\max\{\frac{1}{n},x_m(t)\})
 \frac{g(\max\{\frac{1}{n},x_m(t)\})}{h(\max\{\frac{1}{n},x_m(t)\})}\\
&\leq k(t)h(\frac{1}{n})\frac{g(M)}{h(M)}, \quad  m\in
\{1,2,\dots\}.\end{aligned} \label{e3.2}
\end{equation}
Since
$$
f_n(t,x_m(t),p(t)x'_m(t))\to f_n(t,x_0(t),p(t)x'_0(t)),\quad
\mbox{as } m\to+\infty,
$$
from \eqref{e3.2}, the Lesbegue Dominated Convergence Theorem guarantees
\begin{align*}
&\lim_{m\to+\infty}\max_{t\in[0,1]}|(A_nx_m)(t)-(A_nx_0)(t)|\\
&=\lim_{m\to+\infty}\max_{t\in[0,1]}
|\int_0^1G_1(t,s)\Phi(s)f_n(s,x_m(s),p(s)x_m'(s))ds\\
&\quad -\int_0^1G_1(t,s)\Phi(s)f_n(s,x_0(s),p(s)x_0'(s))ds|\\
&\leq\lim_{m\to+\infty}\max_{t\in[0,1]}
\int_0^1G_1(t,s)\Phi(s)\Big|f_n(s,x_m(s),p(s)x_m'(s)) \\
&\quad -f_n(s,x_0(s),p(s)x_0'(s))\Big|ds\\
&\leq\lim_{m\to+\infty} u_1(0)v_1(1)\int_0^1p(s)\Phi(s)
\Big|f_n(s,x_m(s),p(s)x_m'(s))\\
&\quad -f_n(s,x_0(s),p(s)x_0'(s))\Big|ds
=0
\end{align*}
and
\begin{align*}
&\lim_{m\to+\infty}\max_{t\in[0,1]}|p(t)(A_nx_m)'(t)-p(t)(A_nx_0)'(t)|\\
&=\lim_{m\to+\infty}\max_{t\in[0,1]}
\Big|-\frac{1}{\rho}\int_0^tv_1(s)p(s)\Phi(s)[f_n(s,x_m(s),p(s){x'_m}(s))\\
&\quad -f_n(s,x_0(s),p(s){x'_0}(s))]ds
 +\frac{\alpha}{\rho}\int_t^1u_1(s)p(s)\Phi(s)[f_n(s,x_m(s),p(s)x_m'(s))\\
&\quad -f_n(s,x_0(s),p(s)x_0'(s))]ds\Big|\\
&\leq
\lim_{m\to+\infty}\frac{K}{\rho}\int_0^1p(s)\Phi(s)
\Big|f_n(s,x_m(s),p(s)x_m'(s))-f_n(s,x_0(s),p(s)x_0'(s))\Big|ds\\
&=0,
\end{align*}
which mean that
$$
\lim_{m\to+\infty}\|A_nx_m-A_nx_0\|=0.
$$
Finally, we show $A_n(B_r\cap P)$ is relatively compact.
Obviously, $B_r$ is an unbounded set in $C^1_p$.
Without loss of generality, we suppose $r>1/n$. Then,
for any $x\in B_r\cap P$, we have
\begin{align*}
\max_{t\in[0,1]}|(A_nx)(t)|
&=\max_{t\in[0,1]}|\int_0^1G_1(t,s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds|\\
&=\max_{t\in[0,1]}\int_0^1G_1(t,s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\\
&\leq\max_{t\in[0,1]}\int_0^1G_1(t,s)\Phi(s)k(s)h(\max\{\frac{1}{n},x(s)\})\frac{g(\max\{\frac{1}{n},x(s)\})}{h(\max\{\frac{1}{n},x(s)\})}ds\\
&\leq\max_{t\in[0,1]}\int_0^1G_1(t,s)\Phi(s)k(s)ds h(\frac{1}{n})\frac{g(r)}{h(r)}\\
&\leq u_1(0)v_1(1)\int_0^1p(s)\Phi(s)k(s)ds
h(\frac{1}{n})\frac{g(r)}{h(r)}
\end{align*}
and
\begin{align*}
\max_{t\in[0,1]}|p(t)(A_nx)'(t)|
&=\max_{t\in[0,1]}|-\frac{1}{\rho}\int_0^tv_1(s)p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\\
&\quad +\frac{\alpha}{\rho}\int_t^1u_1(s)p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds|\\
&\leq\frac{K}{\rho}\int_0^1p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\\
&\leq\frac{K}{\rho}\int_0^1p(s)\Phi(s)k(s)h(\max\{\frac{1}{n},x(s)\})\frac{g(\max\{\frac{1}{n},x(s)\})}{h(\max\{\frac{1}{n},x(s)\})}ds\\
&\leq\frac{K}{\rho}\int_0^1p(s)\Phi(s)k(s)ds h(\frac{1}{n})\frac{g(r)}{h(r)},
\end{align*}
which means that $A_n(B_r\cap P)$ is bounded.
Assume that $t, t'\in[0,1]$. Then, for $x\in B_r\cap P$, we have
\begin{align*}
|(A_nx)(t)-(A_nx)(t')|
&=|\int_0^1G_1(t,s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\\
&\quad -\int_0^1G_1(t',s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds|\\
&\leq\int_0^1|G_1(t,s)-G_1(t',s)|\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\\
&\leq\int_0^1|G_1(t,s)-G_1(t',s)|\Phi(s)k(s)dsh(\frac{1}{n})\frac{g(r)}{h(r)}
\end{align*}
and
\begin{align*}
&|p(t)(A_nx)'(t)-p(t')(A_nx)'(t')|\\
&=|-\frac{1}{\rho}\int_0^tv_1(s)p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\\
&\quad +\frac{1}{\rho}\int_0^{t'}v_1(s)p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\\
&\quad +\frac{\alpha}{\rho}\int_t^1u_1(s)p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\\
&\quad -\frac{\alpha}{\rho}
\int_{t'}^1u_1(s)p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds|\\
&\leq 2\frac{K}{\rho}|\int_t^{t'}p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))|ds\\
&\leq 2\frac{K}{\rho}|\int_t^{t'}p(s)\Phi(s)k(s)ds|h(\frac{1}{n})
\frac{g(r)}{h(r)}.
\end{align*}
Then, for any $\varepsilon>0$, we can choose $\delta>0$ small
enough such that
$$
|(A_nx)(t)-(A_nx)(t')|<\varepsilon,\quad
|p(t)(A_nx)'(t)-p(t')(A_nx)'(t')|<\varepsilon,
$$
for all $x\in B_r\cap P$, $|t-t'|<\delta$, $t,t'\in[0,1]$.
Consequently, $\{(A_n(B_r\cap P))(t)\}$ and $\{p(t)(A_n(B_r\cap P))'(t)\}$
is equicontinuous on $[0,1]$.

Consequently, from Arzela-Ascoli theorem, $A_n(B_r\cap P)$ is
relatively compact. The proof is complete.
\end{proof}

\begin{theorem} \label{thm3.3}
Assume that (H1)--(H4) hold. Then \eqref{e1.1} has at least two positive
solutions.
\end{theorem}

\begin{proof}  From Lemma \ref{lem3.2},  for each $n\in\{1,2,\dots\}$,
$A_n:P\to P$ is continuous operator and for any $r>0$,
$A_n(B_r\cap P)$ is relatively compact.
From (H2), choose $R_0>0$ such that
\begin{equation}
\frac{R_0h(R_0)}{u_1(0)v_1(1)\int_0^{1}p(s)\Phi(s)k(s)h(R_0{\gamma}_1(s))ds
g(R_0)}>1.\label{e3.3}
\end{equation}
Without loss of the generality, suppose
$R_0\geq 1/n$, $n\in\{1,2,\dots\}$. Set
$$
\Omega_1=\{x\in C^1_p|\|x\|_1<R_0\}.
$$
Then for any $x\in\partial\Omega_1\cap P$, one has
\begin{equation}
x\not\leq A_nx,\ \ \, n\in\{1,2,\dots\}.\label{e3.4}
\end{equation}
If there exists an $x_0\in \partial{\Omega_1}\cap P$ such
that $x_0\leq A_nx_0$, obviously,
$\max\{\frac{1}{n},x_0(t)\}\geq x_0(t)\geq \gamma_1(t)R_0$.
Then
\begin{align*}
R_0&=\max_{t\in[0,1]}|x_0(t)|\\
&\leq \max_{t\in[0,1]}
\int_0^{1}G_1(t,s)\Phi(s)f(s,\max\{\frac{1}{n},x_0(s)\},p(s){x'}_0(s))ds\\
&\leq u_1(0)v_1(1)\int_0^{1}p(s)\Phi(s)k(s)g(\max\{\frac{1}{n},x_0(s)\})ds\\
&\leq u_1(0)v_1(1)\int_0^{1}p(s)\Phi(s)k(s)h(\max\{\frac{1}{n},x_0(s)\})
\frac{g(\max\{\frac{1}{n},x_0(s)\})}{h(\max\{\frac{1}{n},x_0(s)\})})ds\\
&\leq u_1(0)v_1(1)\int_0^1p(s)\Phi(s)k(s)h(R_0\gamma_1(s))ds
\frac{g(R_0)}{h(R_0)}
\end{align*}
which implies
$$
\frac{R_0h(R_0)}{u_1(0)v_1(1)\int_0^{1}\Phi(s)k(s)h(R_0\gamma_1(s))dsg(R_0)}
\leq 1.
$$
This contradicts \eqref{e3.3}. Then, \eqref{e3.4} is true. From the proof
of Theorem \ref{thm2.5}, the \eqref{e2.13} is true, which means
\begin{equation}
i(A_n,P\cap \Omega_1,P)=1,\quad  n\in\{1,2,\dots\}.\label{e3.5}
\end{equation}
As a result,  for  $n\in\{1,2,\dots\}$, there exists an
$x_n^{(1)}\in P\cap\Omega_1$ with $A_nx_n^{(1)}=x_n^{(1)}$.
Since $\|x_{n}^{(1)}\|_1\leq R_0$, $n\in\{1, 2,\dots\}$, it is
easy to see that $\{x_{n}^{(1)}(t)\}$ and
$\{p(t){x'}_{n}^{(1)}(t)\}$ are uniformly bounded. Moreover, from
Lemma \ref{lem3.1}, the condition (H4), there exists a $\psi_{R_0}(t)$
such that
\begin{equation}
\begin{aligned}
x_{n}^{(1)}(t)
&=\int_0^1G_1(t,s)\Phi(s)f(s,\max\{\frac{1}{n},x_n^{(1)}(s)\},
  p(s)(x_n^{(1)})'(s))ds\\
&\geq \int_0^{1}G_1(t,s)\Phi(s)\psi_{R_0}(s)ds\\
&\geq \gamma_1(t)k^*,
\end{aligned} \label{e3.6}
\end{equation}
where
$k^*=\max_{t\in[0,1]}\int_0^{1}G_1(t,s)\Phi(s)\psi_{R_0}(s)ds$.
Thus, for any $t'$, $t''\in (0,1)$
and $n\in\{1,2,\dots\}$,
\begin{equation}
\begin{aligned}
&|x_{n}^{(1)}(t')-x_{n}^{(1)}(t'')|\\
&\leq \int_0^{1}|G_1(t',s)-G_1(t'',s)
  |f_n(s,x_{n}^{(1)}(s),p(s)(x_n^{(1)})'(s))ds\\
&\leq\int_0^{1}|G_1(t',s)-G_1(t'',s)
  |\Phi(s)k(s)h(\gamma_1(s)k^*)ds \frac{g(R_0)}{h(R_0)},
\end{aligned} \label{e3.7}
\end{equation}
and
\begin{equation}
\begin{aligned}
&|p(t')(x_{n}^{(1)})'(t')-p(t'')(x_{n}^{(1)})'(t'')|\\
&\leq 2\big|\int_{t'}^{t''}p(s)\Phi(s)f_n(s,x_{n}^{(1)}(s),p(s)(x_n^{(1)})'(s))
  ds\big|\\
&\leq 2\big|\int_{t'}^{t''}p(s)\Phi(s)k(s)h(\gamma_1(s)k^*)ds\big|
\frac{g(R_0)}{h(R_0)},\quad  n\in\{1,2,\dots\}.
\end{aligned} \label{e3.8}
\end{equation}
Consequently, for any $\varepsilon>0$, we can choose
a $\delta>0$ such that
$$
|x_{n}^{(1)}(t')-x_{n}^{(1)}(t'')|<\varepsilon,\quad
|p(t')(x_{n}^{(1)})'(t')-p(t'')(x_{n}^{(1)})'(t'')|<\varepsilon,
$$
for all $n\in\{1,2,\dots\}$, $|t'-t''|<\delta$,
$t',t''\in[0,1]$, which implies that
$\{x_n^{(1)}(t)\}$ and $\{p(t)(x_n^{(1)})'(t)\}$ are
equi-continuous on $[0,1]$.

The Arzela-Ascoli theorem guarantees that  there is a subsequence
$\{x_{n_j}^{(1)}\}$ of $\{x_n^{(1)}\}$ with
$\lim_{j\to+\infty}x_{n_j}^{(1)}=x_0^{(1)}$. From \eqref{e3.6}, we
have
$$
x_0(t)\geq k^*\gamma_1(t),\quad \forall t\in[0,1].
$$
Then, for $t\in(0,1)$, if $j$ is big enough,  we have
\begin{align*}
&|f_{n_j}(t,x_{n_j}^{(1)}(t),p(t)(x_{n_j}^{(1)})'(t))
 -f(t,x_0^{(1)}(t),p(t)(x_0^{(1)})'(t))| \\
&=|f(t,x_{n_j}^{(1)}(t),p(t)(x_{n_j}^{(1)})'(t))
  -f(t,x_0^{(1)}(t),p(t)(x_0^{(1)})'(t))|\to 0,\quad\mbox{as }
j\to+\infty
\end{align*}
and
\begin{equation}
\begin{aligned}
f_{n_j}(t,x_{n_j}^{(1)}(t),p(t)(x_{n_j}^{(1)})'(t))
&=f(t,\max\{\frac{1}{n_j},x_{n_j}^{(1)}(t)\},p(t)(x_{n_j}^{(1)})'(t))\\
&\leq k(t)h(k^*\gamma_1(t))\frac{g(R_0)}{h(R_0)}.
\end{aligned} \label{e3.9}
\end{equation}
Then the Lesbegue Dominated Convergence Theorem
guarantees that
\begin{equation}
\begin{aligned}
x_0^{(1)}(t)
&=\lim_{j\to+\infty}x_{n_j}^{(1)}(t)\\
&=\lim_{j\to+\infty}\int_0^{1}G_1(t,s)\Phi(s)f(s,
  \max\{\frac{1}{n_j},x_{n_j}^{(1)}(s)\},
p(s)(x_{n_j}^{(1)})'(s))ds\\
&=\int_0^{1}G_1(t,s)\Phi(s)f(s,x_0^{(1)}(s),p(s)(x_0^{(1)})'(s))ds.
\end{aligned} \label{e3.10}
\end{equation}
Obviously $\|x_0^{(1)}\|_1\leq R_0$. Thus \eqref{e3.3}
can guarantee $\|x_0^{(1)}\|_1<R_0$.
Let $0<a^*<b^*<1$, and $
0<c^*<\min_{t\in[a^*,b^*]}\gamma_1(t)$. Suppose
$$
N^*=\Big(\min_{t\in[a^*,b^*]}\int_{a^*}^{b^*}
G_1(t,s)\Phi(s)k_1(s)dsc^*\Big)^{-1}+1.
$$
From the condition (H3), there exists an $R'>R$ such that
\begin{equation}
g_1(x)>N^*x, \forall x\geq R'.\label{e3.11}
\end{equation}
Now we define
\begin{equation}
\Omega_2=\{x\in \big|\|x\|_1<\frac{R'}{c^*}\}.\label{e3.12}
\end{equation}
We might as well suppose that $\frac{R'}{c^*}>1$, $R'>1$.
Then we have
\begin{equation}
A_nx\not\leq x\quad \text{for all }
 x\in\partial{\Omega_2}\cap P, \;  n\in\{1,2,\dots\}.\label{e3.13}
\end{equation}
Otherwise, suppose there exists $x_0\in \partial{\Omega_2}\cap P$
with $A_nx_0\leq x_0$. Since $x_0\in \partial(\Omega_2)\cap P$,
\begin{equation}
\min_{t\in[a^*,b^*]}x_0(t)\geq\min_{t\in[a^*,b^*]}\gamma_1(t)\|x\|_1
>c^*\frac{R'}{c^*}=R'>1.\label{e3.14}
\end{equation}
 Then, for $t\in[a^*, b^*]$, from \eqref{e3.11} and \eqref{e3.14}, we have
\begin{align*}
 x_0(t)
&\geq(A_nx_0)(t)\\
&=\int_0^{1}G_1(t,s)\Phi(s)f_n(s,x_0(s),p(s)x_0'(s))ds\\
&\geq\int_{a^*}^{b^*}G_1(t,s)\Phi(s)f(s,\max\{\frac{1}{n},x_0(s)\},p(s)x_0'(s))ds\\
&\geq\int_{a^*}^{b^*}G_1(t,s)\Phi(s)k_1(s)g_1(\max\{\frac{1}{n},x_0(s)\})ds\\
&>\int_{a^*}^{b^*}G_1(t,s)\Phi(s)k_1(s)N^*x_0(s)ds\\
&>\int_{a^*}^{b^*}G_1(t,s)\Phi(s)k_1(s)dsN^*c^*\frac{R'}{c^*}\\
&>\frac{R'}{c^*}, \quad \forall n\in\{1,2,\dots\}
\end{align*}
which implies $\|x_0\|_1>R'/c^*$. This contradicts to
$x_0\in P\cap\partial\Omega_2$.

 From \eqref{e3.4} and \eqref{e3.13}, the Theorem \ref{thm2.5} guarantees that
$A_n$ has a fixed point $x_{n}^{(2)}\in(\Omega_2-\overline{\Omega}_1)\cap P$,
$n\in\{1,2,\dots\}$.
It is easy to see that
\begin{equation}
x_n^{(2)}(t)\geq\gamma_1(t)\|x_n^{(2)}\|_1\geq\gamma_1(t)R_0,
\quad n\in\{1,2,\dots\}.\label{e3.15}
\end{equation}
By proof similar to \eqref{e3.6},
\eqref{e3.7}, \eqref{e3.8}, we know that $\{x_n^{(2)}\}$ is relatively compact
in $C^1_p$. Then  there exists a subsequence
$\{x_{n_i}^{(2)}\}$ of $\{x_{n}^{(2)}\}$ with
$\lim_{i\to+\infty}x_{n_i}^{(2)}=x_0^{(2)}$. And moreover, by
similar proof as \eqref{e3.10}, we get $x_0^{(2)}(t)$ is a positive
solution to equation \eqref{e1.1} with
$\frac{R'}{c^*}>\|x_0^{(2)}\|_1>R_0$.
Consequently, equation \eqref{e1.1} has at least two different positive
solution $x_0^{(1)}(t)$ and $x_0^{(2)}(t)$. The proof is complete.
\end{proof}


\begin{example} \label{exa3.1} \rm
Now we consider
\begin{equation}
\begin{gathered}
x''+\frac{1}{16}t^{-1/2}(1-t)^{-1/4}(x^{-1/4}
+x^{2})(1+\cos^2(x'))=0, \quad t\in(0,1)\\
\lim_{t\to 0+}x'(t)=0=x(1).
\end{gathered}
\label{e3.16}
\end{equation}
Then \eqref{e3.16} has at least two positive solutions.
\end{example}

To prove that \eqref{e3.16} has at least two positive solutions, we apply
Theorem \ref{thm3.3} with
$\Phi(t)=\frac{1}{16}t^{-1/2}(1-t)^{-1/4}$,
$p(t)\equiv 1$, $f(t,x,z)=(x^{-1/4}+x^{2})(1+\cos^2(z))$,
$k(t)\equiv 1$, $g(x)=2(x^{-1/4}+x^{2})$,
$h(x)=x^{-1/4}$, $\gamma_1(t)=1-t$, $k_1(t)\equiv 1$,
$g_1(x)=x^{-1/4}+x^{2}$, $\Psi_c(t)=c^{-1/4}$. It
is easy to verify that (H1)--(H4) hold.
Hence, \eqref{e3.16} has at leat two positive solutions.


\begin{example} \label{exa3.2} \rm
 Now we consider
\begin{equation}
\begin{gathered}
x''+\frac{1}{12\pi}t^{-1/4}(1-t)^{-1/4}(x^{-1/4}+x^{2})(\pi+\arctan
{x'})=0, \quad t\in(0,1)\\
x(0)=0=x(1).
\end{gathered} \label{e3.17}
\end{equation}
Then \eqref{e3.17} has at least two positive solutions.
\end{example}

To prove that \eqref{e3.17} has at least two positive solutions, we apply
Theorem \ref{thm3.3} with
$\Phi(t)=\frac{1}{12\pi}t^{-1/4}(1-t)^{-1/4}$,
$p(t)\equiv 1$, $f(t,x,z)=(x^{-1/4}+x^{2})(\pi+\arctan
{x'})$, $k(t)\equiv 1$,
$g(x)=\frac{3\pi}{2}(x^{-1/4}+x^{2})$,
$h(x)=x^{-1/4}$, $\gamma_1(t)=t(1-t)$, $k_1(t)\equiv 1$,
$g_1(x)=x^{-1/4}+x^{2}$, $\Psi_c(t)=c^{-1/4}$. It
is easy to verify that (H1)--(H4) hold.
Hence, \eqref{e3.17} has at leat two positive solutions.


\section{The existence of multiple positive solutions to equation \eqref{e1.2}}

In this section, we consider \eqref{e1.2} and suppose that
$f\in C(R^+\times R^+_0\times R^+,R^+)$,
$p\in C(R^+,R)\cap C(R^+_0,R^+_0)\cap C^1(R^+_0,R)$ with
$\int_0^{+\infty}\frac{1}{p(r)}dr=+\infty$,
$\Phi\in C(R^+_0,R^+)$; here $R^+=[0,+\infty)$, $R^+_0=(0,+\infty)$,
$R=(-\infty,+\infty)$.
Let
$$
G_2(t,s)=\begin{cases} u_2(t)v_2(s)p(s),& a\leq s\leq t<+\infty\\
v_2(t)u_2(s)p(s),& 0\leq t\leq s<+\infty,
\end{cases}
$$
where $u_2(t)=1$ and $ v_2(t)=\int_0^t\frac{1}{p(r)}dr$ for all $t\in R^+$.

Let $ C^1_{\infty}=\big\{x:[0,+\infty)\to R|$ $x(t)$ is
continuous on $R^+$ and $p(t){x'}(t)$ is continuous on $R^+$ also
with $\lim_{t\to+\infty}\frac{x(t)}{1+v_2(t)}$ exists and
$\sup_{t\in[0,+\infty)}p(t)|{x'}(t)|<+\infty\big\}$.
For $x\in C^1_{\infty}$, let
$$
\|x\|_1=\sup_{t\in[0,+\infty)}\frac{|x(t)|}{1+v_2(t)}\quad\text{and}\quad
\|x\|_2=\sup_{t\in[0,+\infty)}p(t)|x'(t)|.
$$
 It is easy to see that $\|\cdot\|_1$ is a norm of $C^1_{\infty}$
and $\|\cdot\|_2$ is a semi-norm of $C^1_{\infty}$. Now Let
$\|x\|=\max\{\|x\|_1,\|x\|_2\}$. Obviously, $C^1_{\infty}$
satisfies (1), (2) and (3) of the Banach space $E$ in section 2.

It is easy to prove that if $x(t)\in C^1_{\infty}$ is a solution
to integral equation
$$
x(t)=\int_0^{\infty}G_2(t,s)\Phi(s)f(s,x(s),p(s)x'(s))ds, \quad
t\in R^+,
$$
then $x(t)$ is a solution to \eqref{e1.2}.

Let $ P=\{x\in C^1_{\infty}|x(t)\geq\gamma_2(t)\|x\|_1,\;
\forall t\in R^+\}$, where
\begin{gather*}
\gamma_2(t)=\begin{cases}
 \int_0^{t}\frac{1}{p(r)}dr, & t\in[0,\tau]\\
 1, & t\in(\tau,+\infty)
\end{cases}
\\
\widetilde{\gamma}_2(t)=\frac{\gamma_2(t)}{ 1+v_2(t)},\quad
t\in R^+;
\end{gather*}
here $\int_0^{\tau}\frac{1}{p(r)}dr=1$.
Suppose that $ x=(1+v_2(t))y$, $t\in R^+$ and
$F(t,y,z)=f(t,(1+v_2(t))y,z)=f(t,x,z)$.

Now we will list some conditions for convenience:
\begin{itemize}
\item[(H1)] There exists a $k\in C(R^+,R^+_0)$,
 a $g\in C(R^+_0,R^+_0)$ and a decreasing continuous function $h\in
C(R^+_0, R^+_0)$ such that
$$
F(t,y,z)\leq k(t)g(y),\quad \forall (y,z)\in R^+_0\times R^+_0,
t\in R^+,
$$
where $ \frac{g(y)}{h(y)}$ is an increasing function and
$\int_0^{\infty}p(s)\Phi(s)k(s)h(c\tilde{\gamma}_2(s))ds<+\infty$
for each $c>0$

\item[(H2)]
$$\sup_{c\in R^+_0}\frac{ch(c)}{\int_0^{\infty}p(s)
\Phi(s)k(s)h(c\widetilde{\gamma}_2(s))ds g(c)}>1
$$

\item[(H3)] There exists a $k_1\in C(R^+,R^+_0)$ and a
$g_1\in C(R^+_0,R^+_0)$ with $F(t,y,z)\geq k_1(t)g_1(y)$, for all
$(t,y,z)\in[0,+\infty)\times R^+_0\times R^+$ such that
$$
\lim_{y\to+\infty}\frac{g_1(y)}{y}=+\infty,
$$
where
$ \int_0^{\infty}p(s)\Phi(s)k_1(s)ds<+\infty$

\item[(H4)] for any  $c>0$,  there exists a $\psi_c\in C(R^+,R^+_0)$
such that $F(t,y,z)\geq \psi_{c}(t)$ for all $(t,y,z)\in R^+\times
(0,c]\times R^+$ with $\int_0^{\infty}p(s)\Phi(s)\psi_c(s)ds<+\infty$.

\end{itemize}
Let $ C_{l}=\big\{x:R^+\to R|$ $x(t)$ is continuous on $R^+$ and
$\lim_{t\to+\infty}x(t)$ exists $\big\}$ with norm
$\|x\|_l=\sup_{t\in[0,+\infty)}|x(t)|$. From \cite{m1}, we know that
$C_{l}$ is a Banach space and following theorem is true.

\begin{theorem}[\cite{m1}] \label{thm4.1}
 Let $M\subseteq C_l(R^+,R)$. Then
$M$ is relatively compact in the space $C_l(R^+,R)$ if the following
conditions hold:
\begin{itemize}
\item[(a)] $M$ is bounded in $C_l$

\item[(b)] the functions belonging to $M$ are locally equi-continuous on
$R^+$;

\item[(c)] the functions from $M$ are equiconvergent, that is, given
$\varepsilon>0$, there corresponds $T(\varepsilon)>0$ such that
$|x(t)-x(+\infty)|<\varepsilon$ for any $t\geq T(\varepsilon)$ and
$x\in M$.
\end{itemize}
\end{theorem}

\begin{theorem}[\cite{o4}] \label{thm4.2}
Let $M\subseteq C_{\infty}^1(R^+,R)$. Then $M$ is relatively compact in
$C_{\infty}^1(R^+,R)$ if the following conditions hold:
\begin{itemize}
\item[(a)] $M$ is bounded in $C_{\infty}^1$;

\item[(b)] the functions belonging to
$\{y\big|y(t)=\frac{x(t)}{1+v_2(t)}, x\in M\}$  and the
functions belonging to $\{y\big|y(t)=p(t)x'(t), x\in M\}$ are
locally equi-continuous on $R^+$;

\item[(c)] the functions from $\{y\big| y(t)=\frac{x(t)}{1+v_2(t)},
x\in M\}$ and the functions from $\{y\big| y(t)=p(t)x'(t),
x\in M\}$ are equi-convergent at $+\infty$.
\end{itemize}
\end{theorem}

\begin{lemma}[\cite{o4}] \label{lem4.3}
 Assume that $\overline{\Phi}(t)\in C(R^+_0,R^+)$ with
$\int_0^{+\infty}p(s)\overline{\Phi}(t)dt<+\infty$ and
$F(t)=\int_0^{\infty}G_2(t,s)\overline{\Phi}(s)ds$. Then
$$
F(t)\geq\gamma_2(t)\frac{F(\tau)}{1+v_2(\tau)},\quad \forall t\in R^+,
\; \tau\in R^+,
$$
and
$$
F(t)\geq\gamma_2(t)\|F\|_1,\quad \forall t\in R^+,\;
\lim_{t\to+\infty}\frac{F(t)}{1+v_2(t)}=0.
$$
\end{lemma}

Let
$f_n(t,x,z)=f(t,\max\{\frac{1}{n}(1+v_2(t)),x\},z), \;
n\in \{1,2,\dots\}$
and for $x\in P$, $n\in\{1,2,\dots\}$, $t\in R^+$, define
\begin{equation}
(A_nx)(t)=\int_0^{\infty}G_2(t,s)\Phi(s)f_n(s,x(s),p(s)x'(s))ds\,.
\label{e4.1}
\end{equation}

\begin{lemma} \label{lem4.4}
Assume that the condition (H1) holds. Then
 for each $n\in\{1,2,\dots\}$, $A_n:P\to P$ is continuous and
for any $r>0$ and $ B_r=\{x\in C^1_{\infty}|\|x\|_1\leq r\}$,
$A_n(P\cap B_r)$ is relatively compact for each $n\geq 1$.
\end{lemma}

\begin{proof}  First, we show that $A_nP\subseteq P$. For any $x\in
P$, we have
\begin{align*}
|(A_nx)(t)|
&=\big|\int_0^{\infty}G_2(t,s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\big|\\
&\leq\int_0^{\infty}G_2(t,s)\Phi(s)f(s,\max\{\frac{1}{n}(1+v_2(s)),x(s)\},
 p(s){x'}(s))ds\\
&=\int_0^{\infty}G_2(t,s)\Phi(s)F(s,\max\{\frac{1}{n},
 \frac{x(s)}{1+v_2(s)}\}, p(s){x'}(s))ds\\
&\leq\int_0^{\infty}G_2(t,s)\Phi(s)k(s)g(\max\{\frac{1}{n},
 \frac{x(s)}{1+v_2(s)}\})ds\\
&\leq\int_0^{\infty}G_2(t,s)\Phi(s)k(s)h(\max\{\frac{1}{n},
 \frac{x(s)}{1+v_2(s)}\})
 \frac{g(\max\{\frac{1}{n},\frac{x(s)}{1+v_2(s)}\})}{h(\max\{\frac{1}{n},
 \frac{x(s)}{1+v_2(s)}\})}ds\\
&\leq\int_0^{\infty}G_2(t,s)\Phi(s)k(s)ds
h(\frac{1}{n})\frac{g(\max\{\frac{1}{n},\|x\|_1\})}{h(\max\{\frac{1}{n},
 \|x\|_1\})}<+\infty
\end{align*}
and
\begin{align*}
|p(t)(A_nx)'(t)|
&=\big|\int_t^{\infty}p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\big|\\
&\leq\int_t^{\infty}p(s)\Phi(s)f(s,\max\{
  \frac{1}{n}(1+v_2(s)),x(s)\},p(s){x'}(s))ds\\
&=\int_t^{\infty}p(s)\Phi(s)F(s,\max\{\frac{1}{n},
  \frac{x(s)}{1+v_2(s)}\},p(s){x'}(s))ds\\
&\leq\int_t^{\infty}p(s)\Phi(s)k(s)g(\max\{\frac{1}{n},
  \frac{x(s)}{1+v_2(s)}\})ds\\
&\leq\int_t^{\infty}p(s)\Phi(s)k(s)h(\max\{\frac{1}{n},
  \frac{x(s)}{1+v_2(s)}\})
  \frac{g(\max\{\frac{1}{n},\frac{x(s)}{1+v_2(s)}\})}{h(\max\{\frac{1}{n},
  \frac{x(s)}{1+v_2(s)}\})}ds\\
&\leq\int_0^{\infty}p(s)\Phi(s)k(s)ds h(\frac{1}{n})
  \frac{g(\max\{\frac{1}{n},\|x\|_1\})}{h(\max\{\frac{1}{n},\|x\|_1\})}\\
  &<+\infty.
\end{align*}
Since $(A_nx)(t)\geq 0$, it is easy to see that $A_n$ is well
defined. Moreover, from Lemma \ref{lem4.3}, for any $x\in P$, we have
$$
(A_nx)(t)\geq \gamma_2(t)\|A_nx\|_1, \quad
\forall t\in R^+.
$$
Consequently, $A_nP\subseteq P$.

Second, we show $A_n:P\to P$ is continuous. Assume that
$\lim_{m\to+\infty}x_m=x_0$, which means there exists an
$M>1/n$ such that $\|x_m\|\leq M$, for all $m\in\{0,1,2,\dots\}$.
Then
\begin{equation}
\begin{aligned}
f_n(t,x_m(t),p(t){x'}_m(t))
&=F(t,\max\{\frac{1}{n},\frac{x_m(t)}{1+v_2(t)}\},p(t)x_m'(t))\\
&\leq k(t)g(\max\{\frac{1}{n},\frac{x_m(t)}{1+v_2(t)}\})\\
&\leq k(t)h(\max\{\frac{1}{n},\frac{x_m(t)}{1+v_2(t)}\})
  \frac{g(\max\{\frac{1}{n},\frac{x_m(t)}{1+v_2(t)}\})}
  {h(\max\{\frac{1}{n},\frac{x_m(t)}{1+v_2(t)}\})} \\
&\leq k(t)h(\frac{1}{n})\frac{g(M)}{h(M)},\quad
m\in\{1,2,\dots\}.
\end{aligned} \label{e4.2}
\end{equation}
Since
$$
 f_n(t,x_m(t),p(t)x'_m(t))\to f_n(t,x_0(t),p(t)x'_0(t)),\quad
 \text{as }m\to+\infty,
$$
from \eqref{e4.2}, the Lesbegue Dominated Convergence Theorem guarantees
that
\begin{align*}
&\lim_{m\to+\infty}\sup_{t\in[0,+\infty)}
  \frac{|(A_nx_m)(t)-(A_nx_0)(t)|}{1+v_2(t)}\\
&=\lim_{m\to+\infty}\sup_{t\in[0,+\infty)}
\big|\int_0^{\infty}\frac{G_2(t,s)}{1+v_2(t)}\Phi(s)(f_n(s,x_m(s),p(s)x_m'(s))\\
&\quad -f_n(s,x_0(s),p(s)x_0'(s)))ds\Big|\\
&\leq \lim_{m\to+\infty}\sup_{t\in[0,+\infty)}
\int_0^{\infty}\frac{G_2(t,s)}{1+v_2(t)}\Phi(s)|f_n(s,x_m(s),p(s)x_m'(s))\\
&\quad -f_n(s,x_0(s),p(s)x_0'(s))|ds\\
&=\lim_{m\to+\infty}\int_0^{\infty}p(s)\Phi(s)|f_n(s,x_m(s),p(s)x_m'(s))\\
 &\quad -f_n(s,x_0(s),p(s)x_0'(s))|ds
=0
\end{align*}
and
\begin{align*}
&\lim_{m\to+\infty}\sup_{t\in[0,+\infty)}|p(t)(A_nx_m)'(t)-p(t)(A_nx_0)'(t)|\\
&=\lim_{m\to+\infty}\sup_{t\in[0,+\infty)}
\Big|\int_t^{\infty}p(s)\Phi(s)f_n(s,x_m(s),p(s)x_m'(s))ds \\
&\quad -\int_t^{\infty}p(s)\Phi(s)f_n(s,x_0(s),p(s)x_0'(s))ds\Big|\\
&\leq\lim_{m\to+\infty}\sup_{t\in[0,+\infty)}
\int_t^{\infty}p(s)\Phi(s)|f_n(s,x_m(s),p(s)x_m'(s))\\
&\quad -f_n(s,x_0(s),p(s)x_0'(s))|ds\\
&=\lim_{m\to+\infty}\int_0^{\infty}p(s)\Phi(s)
\big|f_n(s,x_m(s),p(s)x_m'(s))-f_n(s,x_0(s),p(s)x_0'(s))\big|ds
=0,
\end{align*}
which implies
$$
\lim_{m\to+\infty}\|A_nx_m-A_nx_0\|=0.
$$
Finally, we show $A_n(B_r\cap P)$ is relatively compact.
Obviously, $B_r$ is a unbounded set in $C^1_{\infty}$.
Without loss of generality, we suppose $r>1/n$. Then,
for any $x\in B_r\cap P$, we have
\begin{align*}
&\sup_{t\in[0,+\infty)}\frac{|(A_nx)(t)|}{1+v_2(t)}\\
&=\sup_{t\in[0,+\infty)}|\int_0^{\infty}
  \frac{G_2(t,s)}{1+v_2(t)}\Phi(s)f_n(s,x(s),p(s){x'}(s))ds|\\
&=\sup_{t\in[0,+\infty)}\int_0^{\infty}
  \frac{G_2(t,s)}{1+v_2(t)}\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\\
&\leq\sup_{t\in[0,+\infty)}\int_0^{\infty}
  \frac{G_2(t,s)}{1+v_2(t)}\Phi(s)k(s)h(\max\{\frac{1}{n},
  \frac{x(s)}{1+v_2(s)}\})\frac{g(\max\{\frac{1}{n},
  \frac{x(s)}{1+v_2(s)}\})}{h(\max\{\frac{1}{n},
  \frac{x(s)}{1+v_2(s)}\})}ds\\
&\leq\sup_{t\in[0,+\infty)}\int_0^{\infty}
  \frac{G_2(t,s)}{1+v_2(t)}\Phi(s)k(s)ds h(\frac{1}{n})\frac{g(r)}{h(r)}\\
&\leq\int_0^{\infty}p(s)\Phi(s)k(s)ds
h(\frac{1}{n})\frac{g(r)}{h(r)}
\end{align*}
and
\begin{align*}
&\sup_{t\in[0,+\infty)}|p(t)(A_nx)'(t)|\\
&=\sup_{t\in[0,+\infty)}
  |\int_t^{\infty}p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds|\\
&=\sup_{t\in[0,+\infty)}\int_t^{\infty}p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\\
&\leq\sup_{t\in[0,+\infty)}\int_t^{\infty}p(s)\Phi(s)k(s)h(\max
  \{\frac{1}{n},\frac{x(s)}{1+v_2(s)}\})
\frac{g(\max\{\frac{1}{n},\frac{x(s)}{1+v_2(s)}\})}{h(\max\{\frac{1}{n},
  \frac{x(s)}{1+v_2(s)}\})}ds\\
&\leq\sup_{t\in[0,+\infty)}\int_t^{\infty}p(s)\Phi(s)k(s)ds h(\frac{1}{n})
  \frac{g(r)}{h(r)}\\
&=\int_0^{\infty}p(s)\Phi(s)k(s)ds h(\frac{1}{n})\frac{g(r)}{h(r)},
\end{align*}
which implies $A_n(P\cap B_r)$ is bounded.
Assume that $t, t'\in R^+$. Then, for $x\in B_r\cap P$, we have
\begin{align*}
&|\frac{(A_nx)(t)}{1+v_2(t)}-\frac{(A_nx)(t')}{1+v_2(t')}|\\
&=|\int_0^{\infty}\frac{G_2(t,s)}{1+v_2(t)}
\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\\
&\quad -\int_0^{\infty}\frac{G_2(t',s)}{1+v_2(t')}\Phi(s)f_n(s,x(s),
 p(s){x'}(s))ds|\\
&\leq \int_0^{\infty}|\frac{G_2(t,s)}{1+v_2(t)}-\frac{G_2(t',s)}{1+v_2(t')}
 |\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\\
&\leq \int_0^{\infty}|\frac{G_2(t,s)}{1+v_2(t)}-\frac{G_2(t',s)}{1+v_2(t')}|
  \Phi(s)k(s)dsh(\frac{1}{n})\frac{g(r)}{h(r)}
\end{align*}
and
\begin{align*}
&|p(t)(A_nx)'(t)-p(t')(A_nx)'(t')|\\
&=\big|\int_t^1p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds
  -\int_{t'}^1p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\big|\\
&=\big|\int_t^{t'}p(s)\Phi(s)f_n(s,x(s),p(s){x'}(s))ds\big|\\
&\leq\big|\int_t^{t'}p(s)\Phi(s)k(s)ds\big|h(\frac{1}{n})\frac{g(r)}{h(r)}.
\end{align*}
For any $\varepsilon>0$, $T>0$, we can choose $\delta>0$ small
enough such that
$$
|\frac{(A_nx)(t)}{1+v_2(t)}-\frac{(A_nx)(t')}{1+v_2(t')}|<\varepsilon,\quad
|p(t)(A_nx)'(t)-p(t')(A_nx)'(t')|<\varepsilon,
$$
for all $x\in B_r\cap P$, $|t-t'|<\delta$, $t,t'\in[0,T]$.
 Consequently,
$\{\frac{(A_n(B_r\cap P))(t)}{1+v_2(t)}\}$ and
$\{p(t)(A_n(B_r\cap P))'(t)\}$ is locally equi-continuous on
$[0,+\infty)$.

Moreover, Lemma \ref{lem4.3} guarantees that
\begin{align*}
&\lim_{t\to+\infty}\sup_{x\in P\cap
B_r}\big|\frac{(Ax)(t)}{1+v_2(t)}\big|\\
&\leq\lim_{t\to+\infty}\int_0^{\infty}\frac{G_2(t,s)}{1+v_2(t)}
 \Phi(s)f_n(s,x(s),p(s)x'(s))ds\\
&\leq\lim_{t\to+\infty}\sup_{x\in P\cap
B_r}\int_0^{\infty}\frac{G_2(t,s)}{1+v_2(t)}\Phi(s)k(s)g(\max\{\frac{1}{n},
 \frac{x(s)}{1+v_2(s)}\})ds\\
&\leq\lim_{t\to+\infty}\frac{\int_0^{\infty}G_2(t,s)\Phi(s)k(s)ds}
 {1+v_2(t)}h(\frac{1}{n})\frac{g(r)}{h(r)}
=0\end{align*}
and
\begin{align*}
\lim_{t\to+\infty}\sup_{x\in P\cap B_r}|p(t)(Ax)'(t)|
&\leq\lim_{t\to+\infty}\int_t^{\infty}p(s)\Phi(s)f_n(s,x(s),p(s)x'(s))ds\\
&\leq\lim_{t\to+\infty}\sup_{x\in P\cap
B_r}\int_t^{\infty}p(s)\Phi(s)k(s)g(\max\{\frac{1}{n},x(s)\})ds\\
&\leq\lim_{t\to+\infty}\int_t^{\infty}p(s)\Phi(s)k(s)dsh(\frac{1}{n})
\frac{g(r)}{h(r)}
=0,
\end{align*}
which imply that the functions belonging to
$\{\frac{(A(P\cap B_r))(t)}{1+v_2(t)}\}$ and the functions belonging to
$\{p(t)(A(P\cap B_r))'(t)\}$ are equi-convergent.

As a result, from Lemma \ref{lem4.3}, $A_n(B_r\cap P)$ is relatively
compact.
The proof is complete.
\end{proof}

\begin{theorem} \label{thm4.5}
Assume that (H1)--(H4) hold. Then \eqref{e1.2} has at least two positive
solutions.
\end{theorem}

\begin{proof}  From Lemma \ref{lem4.4},  for each $n\in\{1,2,\dots\}$,
$A_n:P\to P$ is continuous operator and for any $r>0$,
$A_n(B_r\cap P)$ is relatively compact.
From (H2), choose $R_0>0$ such that
\begin{equation}
\frac{R_0h(R_0)}{\int_0^{\infty}p(s)\Phi(s)k(s)h(R_0\tilde{\gamma}_2(s))ds
g(R_0)}>1.\label{e4.3}
\end{equation}
Without loss of the generality, suppose
$R_0\geq 1/n$. Set
\begin{equation}
\Omega_1=\{x\in C^1_{\infty}|\|x\|_1<R_0\}.\label{e4.4}
\end{equation}
Then for any $x\in\partial\Omega_1\cap P$, one has
\begin{equation}
x\not\leq A_nx,\quad  n\in\{1,2,\dots\}.\label{e4.5}
\end{equation}
If there exists $x_0\in \partial{\Omega_1}\cap P$ such
that $x_0\leq A_nx_0$, obviously,
$\max\{\frac{1}{n},x_0(t)\}\geq x_0(t)\geq \gamma_2(t)R_0$ and
$\frac{\max\{\frac{1}{n},x_0(t)\}}{1+v_2(t)}\geq
\tilde{\gamma}_2(t)R_0$. Then
\begin{align*}
R_0&=\sup_{t\in[0,+\infty)}\frac{|x_0(t)|}{1+v_2(t)}\\
&\leq \sup_{t\in[0,+\infty)}
\int_0^{\infty}\frac{G_2(t,s)}{1+v_2(t)}\Phi(s)f(s,
  \max\{\frac{1}{n},x_0(s)\},p(s){x'}_0(s))ds\\
&\leq \int_0^{\infty}p(s)\Phi(s)k(s)g(\frac{\max\{\frac{1}{n},
  \frac{x_0(s)}{1+v_2(s)}\}}{1+v_2(s)})ds\\
&\leq \int_0^{\infty}p(s)\Phi(s)k(s)h(\max\{\frac{1}{n},
  \frac{x_0(s)}{1+v_2(s)}\})
\frac{g(\max\{\frac{1}{n},\frac{x_0(s)}{1+v_2(s)}\})}{h(
  \max\{\frac{1}{n},\frac{x_0(s)}{1+v_2(s)}\})}ds\\
&\leq \int_0^{\infty}p(s)\Phi(s)k(s)h(R_0\tilde{\gamma}_2(s))ds
 \frac{g(R_0)}{h(R_0)},
\end{align*}
which implies
$$
\frac{R_0h(R_0)}{\int_0^{+\infty}p(s)\Phi(s)k(s)h(R_0\tilde{\gamma}_2(s))ds
g(R_0)}\leq 1.
$$
This contradicts \eqref{e4.3}.  Then \eqref{e4.5} is true. From the proof
of Theorem \ref{thm2.5}, the \eqref{e2.13} is true, which means
\begin{equation}
i(A_n,P\cap \Omega_1,P)=1, \quad n\in\{1,2,\dots\}.\label{e4.6}
\end{equation}
So for any $n\geq 1$, there exists an $x_n^{(1)}\in P\cap\Omega_1$
with $A_nx_n^{(1)}=x_n^{(1)}$.

 From $\{x_n^{(1)}\}\subseteq\Omega_1\cap P$, the $\{x_{n}^{(1)}\}$
is bounded. From Lemma \ref{lem4.3}, the condition (H4), there exists a
$\psi_{R_0}(t)$ such that
\begin{equation}
\begin{aligned}
x_n^{(1)}(t)
&=\int_0^{\infty}G_2(t,s)\Phi(s)f(s,\max\{\frac{1}{n},x_n^{(1)}(s)\},
  p(s)(x_n^{(1)})'(s))ds\\
&\geq \int_0^{\infty}G_2(t,s)\Phi(s)\psi_{R_0}(s)ds\\
&\geq \gamma_2(t)k^*, \end{aligned} \label{e4.7}
\end{equation}
where
$k^*=\sup_{t\in[0,+\infty)}\int_0^{\infty}\frac{G_2(t,s)}{1+v_2(t)}
\Phi(s)\psi_{R_0}(s)ds$.
 Then, from Lemma \ref{lem4.3}, one has
\begin{equation}
\begin{aligned}
&\lim_{t\to+\infty}\sup_{n\geq 1}\big|\frac{x_n^{(1)}(t)}{1+v_2(t)}\big|\\
&\leq\lim_{t\to+\infty}\int_0^{\infty}\frac{G_2(t,s)}{1+v_2(t)}
  \Phi(s)f_n(s,x(s),p(s)x'(s))ds\\
&\leq\lim_{t\to+\infty}\sup_{n\geq1}
\int_0^{\infty}\frac{G_2(t,s)}{1+v_2(t)}\Phi(s)k(s)g(\max\{\frac{1}{n},
  \frac{x_n^{(1)}(s)}{1+v_2(s)}\})ds\\
&\leq\lim_{t\to+\infty}\frac{\int_0^{\infty}G_2(t,s)\Phi(s)k(s)
  h(k^*\tilde{\gamma}_2(s))ds}{1+v_2(t)}\frac{g(R_0)}{h(R_0)}
 =0\end{aligned}\label{e4.8}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\lim_{t\to+\infty}\sup_{n\geq 1}|p(t)(x_n^{(1)})'(t)|\\
&\leq\lim_{t\to+\infty}\sup_{n\geq1}\int_t^{\infty}p(s)
  \Phi(s)f_n(s,x(s),p(s)x'(s))ds\\
&\leq\lim_{t\to+\infty}\sup_{n\geq1}\int_t^{\infty}p(s)\Phi(s)k(s)
  g(\max\{\frac{1}{n},\frac{x_n^{(1)}(s)}{1+v_2(s)}\})ds\\
&\leq\lim_{t\to+\infty}\int_t^{\infty}p(s)\Phi(s)k(s)h(k^*\tilde{\gamma}_2(s))
  ds\frac{g(R_0)}{h(R_0)}\\
&=0,
\end{aligned} \label{e4.9}
\end{equation}
which imply that the functions
belonging to $\{\frac{x_n^{(1)}(t)}{1+v_2(t)}\}$ and the
functions belonging to $\{p(t)(x_n^{(1)})'(t)\}$ are
equi-convergent.
Moreover, for any $t', t''\in [0,+\infty)$, $n\in\{1,2,\dots\}$,
\begin{equation}
\begin{aligned}
&|\frac{x_{n}^{(1)}(t')}{1+v_2(t')}-\frac{x_{n}^{(1)}(t'')}{1+v_2(t'')}|\\
&\leq\int_0^{\infty}|\frac{G_2(t',s)}{1+v_2(t')}-\frac{G_2(t'',s)}{1+v_2(t'')}
|f_n(s,x_{n}^{(1)}(s),p(s)(x_n^{(1)})'(s))ds\\
&\leq\int_0^{\infty}|\frac{G_2(t',s)}{1+v_2(t')}-\frac{G_2(t'',s)}{1+v_2(t'')}|\Phi(s)k(s)
h(\tilde{\gamma}_2(s)k^*)ds \frac{g(R_0)}{h(R_0)},
\end{aligned} \label{e4.10}
\end{equation}
and
\begin{equation}
\begin{aligned}
& |p(t')(x_{n}^{(1)})'(t')-p(t'')(x_{n}^{(1)})'(t'')|\\
&\leq\big|\int_{t'}^{t''}p(s)\Phi(s)f_n(s,x_{n}^{(1)}(s),
  p(s)(x_n^{(1)})'(s))ds\big|\\
&\leq\big|\int_{t'}^{t''}p(s)\Phi(s)k(s)h(\tilde{\gamma}_2(s)k^*)ds
  \big|\frac{g(R_0)}{h(R_0)},\quad n\in\{1,2,\dots\}.
\end{aligned} \label{e4.11}
\end{equation}
 Consequently, for any $\varepsilon>0$, $T>0$, we can
choose a $\delta>0$ such that if $|t'-t''|<\delta$
and $t'$, $t''\in [0,T]$, then
$$
|\frac{x_{n}^{(1)}(t')}{1+v_2(t')}-\frac{x_{n}^{(1)}(t'')}{1+v_2(t'')}|
<\varepsilon,\quad
|p(t')(x_{n}^{(1)})'(t')-p(t'')(x_{n}^{(1)})'(t'')|<\varepsilon,
$$
for all $n\in\{1,2,\dots\}$, which implies  that
$\{\frac{x_n^{(1)}(t)}{1+v_2(t)}\}$ and
$\{p(t)(x_n^{(1)})'(t)\}$ are locally equi-continuous on
$[0,+\infty)$.
Thus, Theorem \ref{thm4.2} guarantees that there is a convergent
subsequence $\{x_{n_j}^{(1)}\}$ of $\{x_n^{(1)}\}$ with
$\lim_{j\to+\infty}x_{n_j}^{(1)}=x_0^{(1)}$. From \eqref{e4.7}, we
have
$$
x_0^{(1)}(t)\geq k^*\gamma_2(t),\quad \forall t\in[0,+\infty).
$$
Then, for $t\in(0,+\infty)$, if $j$ is big enough,  we have
\begin{align*}
&|f_{n_j}(t,x_{n_j}^{(1)}(t),p(t)(x_{n_j}^{(1)})'(t))
-f(t,x_0^{(1)}(t),p(t)(x_0^{(1)})'(t))|\\
&=|f(t,x_{n_j}^{(1)}(t),p(t)(x_{n_j}^{(1)})'(t))
-f(t,x_0^{(1)}(t),p(t)(x_0^{(1)})'(t))| \to 0,
\end{align*}
as $j\to+\infty$,  and
\begin{align*}
f_{n_j}(t,x_{n_j}^{(1)}(t),p(t)(x_{n_j}^{(1)})'(t))
&=f(t,\max\{\frac{1}{n_j}(1+v_2(t)),x_{n_j}^{(1)}(t)\},
 p(t)(x_{n_j}^{(1)})'(t))\\
&\leq k(t)h(k^*\tilde{\gamma}_2(t))\frac{g(R_0)}{h(R_0)}.
\end{align*}
Then the Lesbegue Dominated Convergence Theorem guarantees that
\begin{equation}
\begin{aligned}
& x_0^{(1)}(t)\\
&=\lim_{j\to+\infty}x_{n_j}^{(1)}(t)\\
&=\lim_{j\to+\infty}\int_0^{\infty}G_2(t,s)\Phi(s)f(s,
 \max\{\frac{1}{n_j}(1+v_2(s)),x_{n_j}^{(1)}(s)\}, p(s)(x_{n_j}^{(1)})'(s))ds\\
&=\int_0^{\infty}G_2(t,s)\Phi(s)f(s,x_0^{(1)}(s),p(s)(x_0^{(1)})'(s))ds,
\end{aligned} \label{e4.12}
\end{equation}
which implies $x_0^{(1)}(t)$ is a positive solution
to  \eqref{e1.2}. Obviously, $\|x_0^{(1)}\|_1\leq R_0$. Then \eqref{e4.3}
can guarantee $\|x_0^{(1)}\|_1<R_0$.

Let $\tau<a^*<b^*<+\infty$, and $0<c^*<\min_{t\in[a^*,b^*]}\tilde{\gamma}_2(t)$.
 Suppose
$$
N^*=\Big(\min_{t\in[a^*,b^*]}\int_{a^*}^{b^*}
\frac{G_2(t,s)}{1+v_2(t)}\Phi(s)k_1(s)dsc^*\Big)^{-1}+1.
$$
 From condition (H2), there exists an $R'>R$ such that
\begin{equation}
g_1(y)>N^*y, \quad \forall y\geq R'.\label{e4.13}
\end{equation}
Now we define
\begin{equation}
\Omega_2=\{x\in C^1_{\infty}\big|\|x\|_1<\frac{R'}{c^*}\}.\label{e4.14}
\end{equation}
We might as well suppose that $\frac{R'}{c^*}>1$, $R'>1$. Then we
have
\begin{equation}
A_nx\not\leq x, \quad n\in\{1,2,\dots\}\label{e4.15}
\end{equation}
for all $x\in\partial{\Omega_2}\cap P$.
Otherwise, suppose there exists $x_0\in\partial{\Omega_2}\cap P$
with $A_nx_0\leq x_0$. Since $x_0\in \partial(\Omega_2)\cap P$,
$$
\min_{t\in[a^*,b^*]}\frac{x_0(t)}{1+v_2(t)}
 \geq\min_{t\in[a^*,b^*]}\tilde{\gamma}_2(t)
\sup_{t\in[0,+\infty)}\frac{|x(t)|}{1+v_2(t)}>c^*\frac{R'}{c^*}=R'>1.
$$
 Then for $t\in[a^*, b^*]$, from \eqref{e4.13},
 we have
\begin{align*}
\frac{x_0(t)}{1+v_2(t)}&\geq\frac{(A_nx_0)(t)}{1+v_2(t)}\\
&=\int_0^{+\infty}\frac{G_2(t,s)}{1+v_2(t)}\Phi(s)f_n(s,x_0(s),p(s)x_0'(s))ds\\
&\geq\int_{a^*}^{b^*}\frac{G_2(t,s)}{1+v_2(t)}\Phi(s)f(s,
  \max\{\frac{1}{n}(1+v_2(s)),x_0(s)\},p(s)x_0'(s))ds\\
&\geq\int_{a^*}^{b^*}\frac{G_2(t,s)}{1+v_2(t)}
  \Phi(s)k_1(s)g_1(\max\{\frac{1}{n},\frac{x_0(s)}{1+v_2(s)}\})ds\\
&>\int_{a^*}^{b^*}\frac{G_2(t,s)}{1+v_2(t)}\Phi(s)k_1(s)N^*
  \frac{x_0(s)}{1+v_2(s)}ds\\
&>\int_{a^*}^{b^*}G_2(t,s)\Phi(s)k_1(s)dsN^*c^*\frac{R'}{c^*}\\
&>\frac{R'}{c^*},
\end{align*}
which implies $\|x_0\|_1>R'/c^*$. This
contradicts to $x_0\in P\cap\partial\Omega_2$. Then \eqref{e4.15} is
true.

 From \eqref{e4.5} and \eqref{e4.15},  Theorem \ref{thm2.5} guarantees that $A_n$ has a
fixed point $x_{n}^{(2)}\in(\Omega_2-\overline{\Omega}_1)\cap P$.
For the set $\{x_n^{(2)}\}$, since
$\|x_n^{(2)}\|_1=\sup_{t\in[0,+\infty)}\frac{|x_n^{(2)}(t)|}{1+v_2(t)}\leq
\frac{R'}{c^*}$, (H4) guarantees that there is a
$\psi_{R'/c^*}(t)$ such that
\begin{equation}
f(t,x,z)=F(t,y,z)\geq \psi_{\frac{R'}{c^*}}(t), \quad
\forall (t,y,z)\in[0,+\infty)\times(0,\frac{R'}{c^*}]\times[0,+\infty).
\label{e4.16}
\end{equation}
By  proof as in \eqref{e4.7}, \eqref{e4.8}, \eqref{e4.9}, \eqref{e4.10}
and \eqref{e4.11}, we can prove that $\{x_n^{(2)}\}$ is relatively compact in
$C^1_{\infty}$, which means that there exists a subsequence
$\{x_{n_i}^{(2)}\}$ of $\{x_{n}^{(2)}\}$ with
$\lim_{i\to+\infty}x_{n_i}^{(2)}=x_0^{(2)}$. By
proof as in \eqref{e4.12} $x_0^{(2)}(t)$ is a positive solution to
equation \eqref{e1.2} with $\frac{R'}{c^*}>\|x_0^{(2)}\|_1>R_0$.

Consequently, equation \eqref{e1.2} has at least two different positive
solutions $x_0^{(1)}(t)$ and $x_0^{(2)}(t)$. The proof is
complete.
\end{proof}

\begin{example} \label{exa4.1} \rm
Now we consider
\begin{equation}
\begin{gathered}
x''+\frac{1}{16}e^{-t}t^{-1/4}((1+t)^{1/2}
x^{-1/2}+\frac{1}{(1+t)^3}x^3)(1+\frac{{x'}^2}{1+{x'}^2})=0,\;
t\in(0,+\infty)\\
 x(0)=0,\quad  \lim_{t\to+\infty}x'(t)=0.
\end{gathered}
\label{e4.17}
\end{equation}
Then, equation \eqref{e4.17} has at least
two positive solutions.
\end{example}

To prove that \eqref{e4.17} has at least two positive solutions, we apply
Theorem \ref{thm4.5} with $\Phi(t)=\frac{1}{16}e^{-t}t^{-1/4}$,
$p(t)\equiv 1$, $ f(t,x,z)=((1+t)^{1/2}
x^{-1/2}+\frac{1}{(1+t)^3}x^3)(1+\frac{{x'}^2}{1+{x'}^2})$,
$k(t)\equiv 1$, $g(y)=2(y^{-1/2}+y^3)$,
$h(x)=y^{-1/2}$,
$\gamma_1(t)=\begin{cases}
t,&t\in[0,1]\\
1,&t\in(0,+\infty)
\end{cases}$, $\tilde{\gamma}_2(t)=\frac{\gamma_2(t)}{1+t}$,
$k_1(t)\equiv 1$, $g_1(y)=y^{-1/4}+y^{3}$,
$\Psi_c(t)=c^{-1/4}$. It is easy to verify that
(H1)--(H4) hold. Hence, \eqref{e4.17} has at leat two
positive solutions.


\subsection*{Acknowledgements}
 The author wants to thanks Professor Temoney for his guidance,
and the anonymous referee for his/her suggestions.


\begin{thebibliography}{00}

 \bibitem{a1} R. P. Agarwal,  D. O'Regan;
\emph{Nonlinear superlinear singular and nonsingular second order
boundary value problems},
 Journal of Differential Equations, 143 (1998), 60-95.


\bibitem{a2} R. P. Agarwal,  D. O'Regan,  V. Lakshmikantham,  S. Leela;
\emph{Existence of positive
solutions for singular initial and boundary value problems via the
classical upper and lower solution approach},  Nonlinear
Analysis,  50 (2002),   215-222.

\bibitem{a3} R. P. Agarwal,  D. O'Regan;
\emph{Singular Problems on the
Infinite Interval Modelling Phenomena in Draining Flows}, IMA
Journal of Applied Mathematics,  66 (2001),   621-635.

 \bibitem{b1} J. V. Baxley;
\emph{Existence and uniqueness for nonlinear boundary value
problems on infinite intervals}, J. Math. Anal. Appl.,
vol. 147 (1990),  127-133.


 \bibitem{b2} L. E. Bobisud;  \emph{Existence of solutions for nonlinear
singular boundary value problems}, Appl. Anal., vol. 35 (1990), 43-57.


 \bibitem{b3} L. E. Bobisud;
\emph{Existence of positive solutions to some nonlinear singular
boundary value problems on finite and infinite intervals}, J. Math.
Anal. Appl.,  vol. 173  (1993),  69-83.

 \bibitem{b4} L. E. Bosbisud,  D. O'Regan,  W. D. Royalty;
 \emph{Solvability of some
nonlinear boundary value problems},  Nanlinear Anal.,
vol. 12 (1988), 855-869.

  \bibitem{c1} Shaozhu Chen,  Yong Zhang; \emph{Singular
boundary value problems on a half-line}, J. Math. Anal. Appl.,
vol.  195  (1995),  449-468.

 \bibitem{d1} K. Deimling;
 \emph{Nonlinear Functional Analysis}, Springer-Verlag, New York, 1985.

 \bibitem{d2} Dugundji; \emph{An extension of Tietze's theorem},
  Pacific J. Math., vol.  1 (1951), 353-367.

  \bibitem{g1} A. Granas,  R. B. Guenther,  J. W. Lee and D. O'Regan;
\emph{Boundary value problems on infinite intervals and semiconductor
devices},  J. Math. Anal. Appl.,  vol. 116 (1986), 335-345.

 \bibitem{g2} A. Granas, J. Dugudji, \emph{Fixed point Theory}, Springer, 2003.

 \bibitem{g3} Dajun Guo, \emph{Nonlinear Functional Analysis},
 Shandong Science and Technique Press, Jinan, 1985.

 \bibitem{g4} Dajun Guo and V. Lakshmikantham;
\emph{Nonlinear problems in abstract
cones}, Academic Press. Inc. New York, 1988.

 \bibitem{h1} J. Henderson; H. B. Thompson
\emph{Existence of multiple solutions for second order boundary
value problems}, Journal of Differential Equations, vol.  166
(2000), 444-454.


 \bibitem{k1} A. G. Kartsatos, \emph{On the perturbation theorem of
 $m$-accretive operators in Banach spaces}, Proceeding of American Math.
Society, vol. 124 no. 6 (1996), 1811-1820.

 \bibitem{l1} N. G. Lioyd, \emph{Degree theory}, Cambridge Univ.Press,
 Cambridge, 1978.

 \bibitem{m1} M. Meehan,  D. O'Regan;  \emph{Existence theory for
nonlinear Fredholm and Volterra Integral Equations on Half-Open
Intervals},  Nonlinear Anal. ,  vol.  35 (1999), 355-387.

 \bibitem{o1} D. O'Regan,
  \emph{Positive solutions for a class of boundary value problems
on infinite intervals},  Nonlinear Diff. Eqns. Appl., vol. 1
 (1994), 203-228.

 \bibitem{o2} D. O'Regan,  \emph{Theory of Singular
Boundary Value Problems}, World Scientific, Singapore, 1994.


 \bibitem{o3} D. O'Regan, \emph{Existence
theory for ordinary differential equations},
Kluwer,  Dordrecht,  1997.

 \bibitem{o4} D. O'Regan, Baoqiang Yan \& R.P.Agarwal;
\emph{Nonlinear boundary value problems on the semi-infinite
intervals: an upper and lower solution approach}, (to appear)

 \bibitem{p1} P. K. Palamides and G. N. Galanis;
\emph{Positive, unbounded and monotone solutions of the singular
second Painlev equation on the half-line},  Nonlinear Analysis, vol.
57 (2004), 401-419.


 \bibitem{y1} I. Yermachenko, F.Sadyrbaev;
\emph{Types of solutions and multiplicity results for two-point
nonlinear boundary-value problems},  Nonlinear Analysis, in press.

 \bibitem{z1} G. Zhang, C. Wang, Y. Liu;
\emph{Fixed point theorems on unbounded sets},
Journal of Northeast University, vol.  22 No. 2 (2001),
226-228. (in Chinese)

\end{thebibliography}
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