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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 11, pp. 1--11.\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 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/11\hfil Existence and uniqueness of positive solutions]
{Existence and uniqueness of positive solutions for a
 BVP with a p-Laplacian on the half-line}

\author[Y. Tian, W. Ge\hfil EJDE-2009/11\hfilneg]
{Yu Tian, Weigao Ge}

\address{Yu Tian \newline
 School of Science, Beijing University of Posts and
Telecommunications, Beijing 100876,  China}
\email{tianyu2992@163.com}

\address{Weigao Ge \newline
Department of Applied Mathematics, Beijing Institute
of Technology,  Beijing 100081, China}

\thanks{Submitted May 23, 2008. Published January 9, 2009.}
\thanks{Supported by grants: 10671012 from the National Natural
Sciences Foundation of China, \hfill\break\indent
 20050007011 from Foundation for PhD Specialities of Educational
 Department of China, \hfill\break\indent
 10726038 from Tianyuan Fund of Mathematics in China}
\subjclass[2000]{34B10, 34B18, 34B40}
\keywords{Multi-point boundary-value problem;
p-Laplacian; half-line; \hfill\break\indent
 positive solutions; existence; uniqueness}

\begin{abstract}
 In this work, we consider the second order multi-point
 boundary-value problem  with a p-Laplacian
 \begin{gather*}
 (\rho(t)\Phi_p(x'(t)))'+f(t, x(t), x'(t))=0,\quad t\in [0,+\infty),\\
 x(0)=\sum_{i=1}^{m}\alpha_i x(\xi_i), \quad
 \lim_{t\to\infty}x(t)=0\,.
 \end{gather*}
 By applying a nonlinear alternative theorem,
 we establish existence and uniqueness of solutions on the half-line.
 Also a uniqueness result for positive solutions is  discussed
 when $f$ depends on the first-order derivative. The emphasis here
 is on the one dimensional p-Laplacian operator.
\end{abstract}

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

\section{Introduction}

  In recent years, a great deal of work has been done
in the study of multi-point boundary-value problems which arise in
different areas of applied mathematics and physics. The study of
multi-point boundary-value problems for linear second order
differential equations was initiated by Il'in and Moiseev
\cite{duodian}. Since then, more general nonlinear multi-point
boundary-value problems were studied by several authors, see
\cite{duzengji,hege,madexiang,maruyun,tian1}
and the references cited therein.


For a finite interval,  He and Ge \cite{hege} used the Leggett-Williams fixed
point theorem to the following second-order three-point boundary-value problem
\begin{equation}\label{000}
\begin{gathered}
u''(t)+f(t, u(t)) = 0, \quad t\in (0, 1),\\
  u(0)=0,\quad u(1)=\xi u(\eta),
\end{gathered}
\end{equation}
where $\xi>0$, $0<\eta< 1 $ and $\xi\eta<1$.
 Du, Xue and Ge \cite{duzengji} applied  Leray-Schauder degree theory
and  lower and
upper solutions method to \eqref{000} when  $f$ does not depend on
the first-order derivative explicitly, and obtained the existence of
at least three solutions.  Nonlinear differential equation on finite
interval
\begin{equation}\label{equation}
(\Phi_p(u'))'+ f (t, u, u')=0,\quad t\in (0, 1)
\end{equation}
with different boundary conditions has been studied extensively.
 Ma, Du and Ge \cite{madexiang} obtained some criteria for the existence
of monotone positive solutions to the equation \eqref{equation} with
boundary condition $u'(0)
=\sum_{i=1}^{n}\alpha_iu'(\xi_i),\quad u(1)=
\sum_{i=1}^{n}\beta_i u(\xi_i)$.

For a infinite interval, in a monograph \cite{agarwalzhuanzhuinfinite}
Agarwal and  O'Regan  studied two-point boundary-value problems on
the half-line and obtained a series of  interesting results.
Inspired by \cite{agarwalzhuanzhuinfinite}, many authors devoted
the study of two-point and multi-point  boundary-value problems on
the half-line, see
\cite{baichuanzhi,tian1,yanbaoqiang,yan,zima}.
Tian and Ge \cite{tian1}
established the existence of at least three positive solutions for
the  problem
\begin{equation}\label{eqm0}
\begin{gathered}
(\rho(t)x'(t))'+f(t, x(t), x'(t))=0,\quad t\in I=[0,+\infty),\\
x(0)=\alpha x(\xi), \quad \lim_{t\to\infty}x(t)=0,
\end{gathered}
\end{equation}
where $\rho\in C[0, +\infty)\cap C^1(0, +\infty), \rho(t)>0$ for
$t\in [0, +\infty)$, $\int_0^{\infty}\frac{1}{\rho(t)}dt<\infty$,
$\alpha \ge0$, $0\le \xi<\infty$, $f: I\times I\times R\to I$.

However, in   \cite{duzengji,hege,maruyun,tian1}, the
one dimensional p-Laplacian operator is not involved.
Ma, Du and Ge \cite{madexiang}  studied only
boundary-value problem on finite interval and the nonlinear term
does not depend
on the first order derivative explicitly. Moreover, only existence
results were established in the above literature. By so far, very
few existence and uniqueness results were established for
multi-point boundary-value problem with a p-Laplacian on the
half-line.

 Motivated by the above
results, we consider the existence of positive solutions for
multi-point boundary-value problem
\begin{equation}\label{eqm}
\begin{gathered}
(\rho(t)\Phi_p(x'(t)))'+f(t, x(t), x'(t))=0,\quad t\in I=[0,
+\infty),\\
x(0)=\sum_{i=1}^{m}\alpha_i x(\xi_i), \quad
\lim_{t\to\infty}x(t)=0.
\end{gathered}
\end{equation}
where $\Phi_p(s)=|s|^{p-2}s$, $p>1$, $\xi_i\in (0, \infty)$,
$i=1, 2, \dots, m$, and $\alpha_i, \rho, f$ satisfy
\begin{itemize}

\item[(H1)] $0\le \alpha_i<1, (i=1, 2, \dots, m) $ satisfies $0\le
\sum_{i=1}^{m}\alpha_i<1$,

\item[(H2)] $\rho\in C[0, +\infty)\cap C^1(0, +\infty), \rho(t)>0$ for
$t\in [0, +\infty)$, and non-decreasing on $[0, +\infty)$,
 $\int_0^{\infty} \Phi_p^{-1} (1/\rho(t))dt<\infty$,

\item[(H3)] $f: [0, +\infty)\times [0, +\infty)\times R\to[0,+\infty)$
is an L$^1$-Carath\'edory function, that is \\
(i) $t\to f(t, x, y)$ is measurable for any $(x, y)\in [0,
 +\infty)\times R$,\\
(ii) $(x, y)\to f(t, x, y)$ is continuous for a.e. $t\in I$,\\
(iii) for each $r_1, r_2>0$ there exists $l_{r_1, r_2}\in L^{1}[0,
\infty)$ such that $|x|\le r_1, |y|\le r_2$ imply $|f(t, x, y)|\le
l_{r_1, r_2}(t)$ for almost all $t\in I$.

\end{itemize}
Furthermore, when $f$ does not depend on the first-order derivative
explicitly, we establish the uniqueness result of positive
solutions. We note that when $p=2$, $\xi_1=\xi_2=\dots=\xi_m$,
problem \eqref{eqm}  reduces to \eqref{eqm0}.

\begin{definition} \label{def1.1} \rm
A function $x$ is said to
be a positive solution of boundary-value problem \eqref{eqm}, if
$x\in C^1(I, I), (\Phi_p(x'(t)))'\in L^1(I), x(t)\ge 0, $ and $x$
satisfies \eqref{eqm} for $t\in I$.
\end{definition}

By using fixed point theorem on cone, we establish the existence of
positive solutions for problem \eqref{eqm}. In order to apply fixed
point theory, it is very important to transform BVP into an
equivalent integral equation. For $p=2$, the process  is easy to be
realized since the Green's function exists, however, for $p\neq 2$,
it is impossible since the differential operator $(\Phi_p(x'))'$
is nonlinear. Besides,  nonlinearity $f$ depends on the first-order
derivative, which brings about much trouble, such as, the
verification of the compactness and continuity of the operator.

In this paper, we will need the following lemmas.


\begin{lemma}[Nonlinear alternative \cite{meehan}]
\label{nonlinearalternative}
 Let $C$ be a convex subset of a normed linear space $E$, and $U$
be an open subset of $C$, with $p^*\in U$. Then every compact,
continuous map $N: \overline{U}\to C$ has at least one of the following two
properties:
\begin{itemize}
\item[(a)] $N$ has a fixed point;
\item[(b)] there is an $x\in \partial U$, with
$x=(1-\overline{\lambda})p^*+\overline{\lambda}Nx$ for some
$0< \overline{\lambda}<1$.
\end{itemize}
\end{lemma}

\begin{lemma}[\cite{corduneanu,meehan}] \label{compact}
Let $ C_l([0, \infty), R)=\{x\in C([0, \infty)):
\lim_{t\to\infty}x(t)\text{ exists}\}$, then subset $M$ of $C_L$
is precompact if the
following conditions hold:
\begin{itemize}
\item[(a)] $M$ is bounded in $C_l$;
\item[(b)] the functions belonging to $M$
are locally equicontinuous on any interval of $[0, \infty)$;
\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 $t\ge T(\varepsilon)$ and
$x\in M$.
\end{itemize}
\end{lemma}

\section{Related Lemmas}

 We consider the Banach space $E=\{x\in C^1(I):
\lim_{t\to \infty}x(t)=0\}$ equipped with the norm
\[
\|x\|=\max\{\|x\|_0, \|x'\|_0\}, \quad
\|x\|_0=\sup_{t\in I}|x(t)|.
\]
Let $P=\{x\in E: x(t)\ge0, t\in I\}$.

Let $x\in P$. Suppose that $x$ is a solution of BVP
\begin{equation}\label{m3.1}
\begin{gathered}
(\rho(t)\Phi_p(x'(t)))'+f(t, x(t), x'(t))=0,\quad t\in I=[0,+\infty),\\
x(0)=\sum_{i=1}^{m}\alpha_i x(\xi_i), \quad
\lim_{t\to\infty}x(t)=0.
\end{gathered}
\end{equation}
Then
\begin{gather*}
\Phi_p(x'(t))=\frac{1}{\rho(t)}\Big(\rho(0)A_x-\int_0^{t}f(r,
x(r), x'(r))dr\Big), \\
x(t)=x(0)+\int_0^{t}\Phi_p^{-1}
\Big[\frac{1}{\rho(s)}\Big(\rho(0)A_x-\int_0^{s}f(r,
x(r), x'(r))dr\Big)\Big]ds.
\end{gather*}
Since $x$ satisfies
$x(0)=\sum_{i=1}^{m}\alpha_i x(\xi_i)$, by computing, one has
\[
x(0)=\frac{1}{1-\sum_{i=1}^{m}\alpha_i}\sum_{i=1}^{m}\alpha_i
\int_0^{\xi_i}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r), x'(r))dr\Big)\Big]ds.
\]
Thus
\begin{align*}
x(t)
&=\frac{1}{1-\sum_{i=1}^{m}\alpha_i}\sum_{i=1}^{m}\alpha_i
\int_0^{\xi_i}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r),
x'(r))dr\Big)\Big]ds\\
& \quad +\int_0^{t}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r),
x'(r))dr\Big)\Big]ds.
\end{align*}
The second boundary
condition $\lim_{t\to\infty}x(t)=0$ means that $A_x$
satisfies
\begin{equation}\label{2.1}
\begin{aligned}
&\frac{1}{1-\sum_{i=1}^{m}\alpha_i}\sum_{i=1}^{m}\alpha_i
\int_0^{\xi_i}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r),
x'(r))dr\Big)\Big]ds\\
& +\int_0^{\infty}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r),
x'(r))dr\Big)\Big]ds=0.
\end{aligned}
\end{equation}

\begin{lemma}\label{lm3.1}
For $x\in P$, there exists a unique $A_x\in
\big(0, \frac{1}{\rho(0)}\int_0^{\infty}f(r, x(r),
x'(r))dr\big)$ satisfying \eqref{2.1}.
\end{lemma}

\begin{proof}
Let $x\in P$. Define
\begin{align*}
H_x(c)&= \frac{1}{1-\sum_{i=1}^{m}
\alpha_i}\sum_{i=1}^{m}\alpha_i\int_0^{\xi_i}\Phi_p^{-1}
\Big[\frac{1}{\rho(s)}\Big(\rho(0)c-\int_0^{s}f(r, x(r),
x'(r))dr\Big)\Big]ds\\
&\quad +\int_0^{\infty}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)c-\int_0^{s}f(r, x(r), x'(r))dr\Big)\Big]ds.
\end{align*}
Then $H_x\in C(R, R)$ is increasing  and $H_x(0)<0$. Let
\[
\overline{c}=\frac{1}{\rho(0)}\int_0^{\infty}f(r, x(r), x'(r))dr,
\]
then $H_x(\overline{c})>0$. By mean value theorem,  there
exists  $A_x\in (0, \overline{c})$ satisfying $H_x(A_x)=0$. Since
 $H_x(c)$ is increasing about $c$, there exists a unique $A_x$
satisfying $H_x(A_x)=0$.
 \end{proof}


\begin{lemma}\label{lm2}
The function $A_x: P\to [0, +\infty)$ is continuous on $x$.
\end{lemma}
\begin{proof}
Let $\{x_n\}\in P$ with $x_n\to x_0\in P$ as $n\to\infty$ in $P$.
Let $\{A_{x_n}\}(n=1, 2, \dots, m)$ be  constants decided by
equation \eqref{2.1} corresponding to $x_n(n=1, 2, \dots, m)$.
Since $x_n\to x_0$ in $P$ as $n\to\infty$, there
exists an $M>0$ such that $\|x_n\|\le M.$ The fact $f$ is an
$L^1-$Carath\'edory function means
\[
\int_0^{\infty}|f(r, x_n(r), x_{n}'(r))-f(r, x_0(r), x_0'(r))|dr
\le 2\int_0^{\infty}l_{M,M}(r)dr<\infty;
\]
that is,
 \[
\int_0^{\infty}f(r, x_n(r),
x_{n}'(r))dr\le\int_0^{\infty}f(r, x_0(r), x_0'(r))dr+
2\int_0^{\infty}l_{M,M}(r)dr<\infty.
\]
So
\begin{align*}
A_{x_n}&\in
\Big(0, \frac{1}{\rho(0)}\int_0^{\infty}f(r, x_n(r), x_{n}'(r))dr\Big)\\
&\subseteq \Big(0, \frac{1}{\rho(0)}\Big(\int_0^{\infty}f(r,
x_0(r), x_0'(r))dr+
2\int_0^{\infty}l_{M,M}(r)dr\Big)\Big),
\end{align*}
which means that $\{A_{x_n}\}$ is bounded.

Suppose that $\{A_{x_n}\}$ does not converge to $A_{x_0}$.
Then there exist two subsequences $\{A_{x_{n_k}}^{(1)}\}$ and
$\{A_{x_{n_k}}^{(2)}\}$ of $\{A_{x_{n_k}}\}$ with
$A_{x_{n_k}}^{(1)}\to c_1$ and $A_{x_{n_k}}^{(2)}\to
c_2$ since $\{A_{x_n}\}$ is bounded, but $c_1\neq c_2$. By the
construction of $A_{x_n}$, $(n=1, 2, \dots)$, we have
\begin{align*}
&\frac{1}{1-\sum_{i=1}^{m}\alpha_i}\sum_{i=1}^{m}\alpha_i
\int_0^{\xi_i}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_{x_{n_k}}^{(1)}-\int_0^{s}f(r, x_{n_k}^{(1)}(r),
x_{n_k}^{(1)'}(r))dr\Big)\Big]ds\\
& +\int_0^{\infty}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_{x_{n_k}}^{(1)}-\int_0^{s}f(r, x_{n_k}^{(1)}(r),
x_{n_k}^{(1)'}(r))dr\Big)\Big]ds=0.
\end{align*}
Let $n_k\to\infty$, using Lebesgue's dominated convergence
theorem, the above equality implies
\begin{align*}
 &\frac{1}{1-\sum_{i=1}^{m}
\alpha_i}\sum_{i=1}^{m}\alpha_i\int_0^{\xi_i}\Phi_p^{-1}
\Big[\frac{1}{\rho(s)}\Big(
\rho(0)c_1-\int_0^{s}f(r, x_0(r),
x_0'(r))dr\Big)\Big]ds\\
& +\int_0^{\infty}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)c_1-\int_0^{s}f(r, x_0(r),
x_0'(r))dr\Big)\Big]ds=0.
\end{align*}
Since $\{A_{x_n}\}$ ($n=1, 2, \dots$) is  unique with respect to $x_n$,
we get $c_1=A_{x_0}$. Similarly, $c_2=A_{x_0}$. Thus $c_1=c_2$, a
contradiction. So, for any $x_n\to x_0$, one has
$A_{x_n}\to A_{x_0}$, which means $A_x: P\to R$ is
continuous.
\end{proof}

Define the operator $T$ on $P$ as
\begin{equation}\label{T}
\begin{aligned}
&Tx(t)\\
&=\frac{1}{1-\sum_{i=1}^{m}\alpha_i}\sum_{i=1}^{m}\alpha_i
\int_0^{\xi_i}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r),
x'(r))dr\Big)\Big]ds\\
&\quad  +\int_0^{t}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r), x'(r))dr\Big)\Big]ds,
\end{aligned}
\end{equation}
where $A_x$ is defined in \eqref{2.1} corresponding to $x$. By Lemma
\ref{lm2}, we know that $T$ is well defined. The fixed point
$x\in P$ of the operator $T$ is just a positive solution of
\eqref{eqm}.

\begin{lemma}\label{le2.4}
The operator $T: P\to P$ is completely continuous.
\end{lemma}

\begin{proof}
  (1) First we show that  the operator $T$ maps $P$ to $P$.
 By the construction of $T$, there exists $\tau\in (0, \infty)$ such that
 \begin{equation}\label{*}
        Tx(t) \text{ is increasing for $t\in [0, \tau]$ and decreasing
for $t\in [\tau,\infty)$}.
\end{equation}
 If we show $Tx(0)\ge0$, then $Tx(t)\ge0, t\in I$ since
\eqref{*} and $\lim_{t\to\infty}Tx(t)=0$ hold.
 For this, we assume that $Tx(0)<0$. Since
 $\lim_{t\to\infty}Tx(t)=0$ and \eqref{*} holds,
 there exists $t_0>0$ such that $Tx(t)\ge0, t\in [t_0, \infty)$.
 Without loss of generality, we assume there exists
$i_0\in \{1, 2, \dots,  m\}$ such that
$Tx(\xi_i)<0, i=1, \dots,  i_0$ and
$Tx(\xi_i)\ge0, i=i_0+1, \dots, m$. Then
\[
Tx(0)=\sum_{i=1}^{m}\alpha_i Tx(\xi_i)\ge \sum_{i=1}^{i_0}\alpha_i
Tx(\xi_i)\ge \sum_{i=1}^{i_0}\alpha_i Tx(0).
\]
 So $\sum_{i=1}^{i_0}\alpha_i\ge1$, a contradiction. Thus,
 $Tx(0)\ge 0$ and so $Tx(t)\ge0, t\in I$.


(2) Next we  show that $T$ is continuous on $P$. From the
continuity of $f$ and $A_x$, the result follows.

 (3) Next we show that $T$ is relatively compact.
Given a bounded set $D\subseteq P$. Then, there exists $M>0$ such that
$D\subseteq \{x\in P: \|x\|\le M\}$. For any $x\in D$, we have
 \[
\int_0^{\infty}f(t, x(t), x'(t))dt\le \int_0^{\infty}l_{M, M}(t)dt:=L.
\]
Thus $|A_x|\le\frac{L}{\rho(0)}$. Therefore,
\begin{gather*}
\|Tx\|_0\le
 \Phi_p^{-1}(2L)\frac{\sum_{i=1}^{m} \alpha_i
\int_0^{\xi_i} \Phi_p^{-1} \big(\frac{1}{\rho(s)}\big)ds}{1-\sum_{i=1}^{m}\alpha_i}
+\Phi_p^{-1}(2L)\int_0^{\infty}\Phi_p^{-1}\big(\frac{1}{\rho(s)}\big)ds
<\infty.
\\
\|(Tx)'\|_0\le\Phi_p^{-1}(2L)\sup_{t\in I}\Phi_p^{-1}
\big(\frac{1}{\rho(t)}\big).
\end{gather*}
Since the condition (H2) holds, one has
$\sup_{t\in I}\Phi_p^{-1}\big(\frac{1}{\rho(t)}\big)<\infty$,
which means that $\|(Tx)'\|_0<\infty$.
 So, $\{TD(t)\}$, $\{(TD)'(t)\}$ are bounded. Besides,
$\{TD(t)\}$ is equi-continuous. Now we shall show that $\{(TD)'(t)\}$
is local equi-continuous on $I$.
 For any $K>0$,
 $t_1, t_2\in[0, K]$ and $x\in D$, then
 \begin{align*}
 & |\Phi_p((Tx)'(t_1))-\Phi_p((Tx)'(t_2))|\\
&= \big|\frac{1}{\rho(t_1)}
\Big(\rho(0)A_x-\int_0^{t_1}f(r, x(r), x'(r))dr\Big)\\
&\quad - \frac{1}{\rho(t_2)}\Big(\rho(0)A_x
 -\int_0^{t_2}f(r, x(r), x'(r))dr\Big)
 \big|\\
 &\leq |\frac{1}{\rho(t_1)}-\frac{1}{\rho(t_2)}|\times
|\rho(0)A_x-\int_0^{t_1}f(s, x(s), x'(s))ds|\\
&\quad  +\frac{1}{\rho(t_2)}|\int_{t_1}^{t_2}f(r, x(r), x'(r))dr|\\
 &\leq |\frac{1}{\rho(t_1)}-\frac{1}{\rho(t_2)}|\times 2L
  +\frac{1}{\rho(t_2)}|\int_{t_1}^{t_2}l_{M, M}(r)dr|.
 \end{align*}
Since
 $\int_0^{\infty}\frac{1}{\rho(s)}ds<\infty,
 \int_0^{\infty}l_{M, M}(r)dr<\infty$, for any $\varepsilon>0$, there
 exists $\delta>0$, such that
 $|\Phi_p(Tx)'(t_1)-\Phi_p(Tx)'(t_2)|<\varepsilon$ for any
 $|t_1-t_2|<\delta$. Noticing $\Phi_p(x)$ is continuous about $x$,
 $|(Tx)'(t_1)-(Tx)'(t_2)|<\varepsilon'$.
 Therefore, $\{(TD)'(t)\}$ is equi-continuous.

(4) At last we will show that  $T$ is equiconvergent at $\infty$.
Since $\lim_{t\to\infty}Tx(t)=0$, one has
\begin{align*}
&\lim_{t\to\infty}|(Tx)(t)-(Tx)(\infty)|\\
&=\lim_{t\to\infty}|(Tx)(t)|\\
&=\lim_{t\to\infty}\Big|
\frac{1}{1-\sum_{i=1}^{m}\alpha_i}\sum_{i=1}^{m}\alpha_i
\int_0^{\xi_i}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r),
x'(r))dr\Big)\Big]ds \\
&\quad +\int_0^{t}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r),
x'(r))dr\Big)\Big]ds\Big|\\
&=\Big| \frac{1}{1-\sum_{i=1}^{m}\alpha_i}\sum_{i=1}^{m}\alpha_i
 \int_0^{\xi_i}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r), x'(r))dr\Big)\Big]ds\\
&\quad +\int_0^{\infty}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r), x'(r))dr\Big)\Big]ds\Big|.
\end{align*}
Since $A_x$ satisfies \eqref{2.1}, one has
\[
\lim_{t\to\infty}|(Tx)(t)-(Tx)(\infty)|=0.
\]
Therefore, $T: P\to P$ is equiconvergent at $\infty$.

By Lemma \ref{compact}, the operator $T: P\to P$ is
completely continuous.
  \end{proof}

 \section{Existence of positive solutions}

 For convenience, we denote
 \begin{gather}\label{de1}
 \Delta_1=\max\Big\{\frac{1}{1-\sum_{i=1}^{m}\alpha_i}\sum_{i=1}^{m}
\alpha_i\int_0^{\xi_i}
 \Phi_p^{-1}\big(\frac{1}{\rho(s)}\big)ds,
 \int_0^{\infty}\Phi_p^{-1}\big(\frac{1}{\rho(s)}\big)ds\Big\}
\\ \label{de2}
\Delta_2=\sup_{t\in
I}\Phi_p^{-1}\big(\frac{1}{\rho(t)}\big).
\end{gather}

 \begin{theorem}\label{thm1}
Suppose that {\rm (H1)--(H3)} hold and $f(t, 0, 0)\not\equiv 0$ for
 $t\in I$. Also assume there exist functions $a, b, c\in L^1([0,
 \infty), [0, \infty))$ satisfying
\[
 \Phi_p^{-1}(\|b\|_{L^1})+ \Phi_p^{-1}(\|c\|_{L^1})
<\min\Big\{\frac{1}{3^{q-1}\Delta_1}, \frac{1}{3^{q-1}\Delta_2}
 \Big\},
\]
where $\frac{1}{p}+\frac{1}{q}=1, \|b\|_{L^1}=\int_0^{\infty}|b(t)|dt$,
such that
\[
f(t, x, y)\le  a(t)+b(t)\Phi_p(x)+c(t)\Phi_p(|y|).
\]
Then problem \eqref{eqm} has at least one nontrivial positive solution.
 \end{theorem}

 \begin{proof}
We will apply Lemma \ref{nonlinearalternative} to
 show this theorem.
 From Lemma \ref{le2.4}, $T: P\to P$ is a completely
 continuous operator.
Let
\begin{align*}
M>\max\Big\{&\frac{3^{q-1}\Delta_1 \Phi_p^{-1}(\|a\|_{L^1})}
 {1-3^{q-1}\Delta_1 (\Phi_p^{-1}(\|b\|_{L^1
 })+\Phi_p^{-1}(\|c\|_{L^1}))},\\
& \frac{3^{q-1}\Delta_2 \Phi_p^{-1}(\|a\|_{L^1})}
 {1-3^{q-1}\Delta_2 (\Phi_p^{-1}(\|b\|_{L^1})+\Phi_p^{-1}(\|c\|_{L^1}))}
\Big\},
\end{align*}
where $\frac{1}{p}+\frac{1}{q}=1$.
 Now we define $\Omega=\{x\in P: \|x\|< M\}$. For any
$x\in \partial\Omega$,  $\|x\|=M$, so
$\|x\|_0\le M, \|x'\|_0\le M$, by assumption of  theorem
and Lemma \ref{lm3.1},
 \begin{align*}
|Tx(t)|
&=\frac{1}{1-\sum_{i=1}^{m}\alpha_i}\sum_{i=1}^{m}\alpha_i
\int_0^{\xi_i}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r),
x'(r))dr\Big)\Big]ds \\
&\quad +\int_0^{t}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r), x'(r))dr\Big)\Big]ds \\
&\leq \frac{1}{1-\sum_{i=1}^{m}\alpha_i}\sum_{i=1}^{m}\alpha_i
 \int_0^{\xi_i}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}
 \int_0^{\infty}f(r, x(r), x'(r))dr\Big]ds\\
&\quad +\int_0^{\infty}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}
 \int_0^{\infty}f(r, x(r), x'(r))dr\Big]ds\\
&\leq \frac{1}{1-\sum_{i=1}^{m}\alpha_i}\sum_{i=1}^{m}\alpha_i
 \int_0^{\xi_i}\Phi_p^{-1}\big(\frac{1}{\rho(s)}\big)ds
 \Phi_p^{-1}\Big(\int_0^{\infty}f(r, x(r), x'(r))dr\Big)\\
&\quad +\int_0^{\infty}\Phi_p^{-1}\big(\frac{1}{\rho(s)}\big)ds
 \Phi_p^{-1}\Big(\int_0^{\infty}f(r, x(r), x'(r))dr\Big)\\
&\leq \Delta_1\Phi_p^{-1}(\|a\|_{L^1}+\|b\|_{L^1}\Phi_p(\|x\|_0)
 +\|c\|_{L^1}\Phi_p(\|x'\|_0)).
\end{align*}
\begin{align*}
 |(Tx)'(t)|&=\Big|\Phi_p^{-1}\Big[\frac{1}{\rho(t)}
\Big(\rho(0)A_x-\int_0^{t}f(r, x(r), x'(r))dr\Big)\Big]\Big|\\
&\leq \sup_{t\in I}\Phi_p^{-1}\big(\frac{1}{\rho(t)}\big)\Phi_p^{-1}
\Big(\int_0^{\infty}f(r,x(r), x'(r))dr\Big)\\
&\leq \Delta_2\Phi_p^{-1}(\|a\|_{L^1}+\|b\|_{L^1}\Phi_p(\|x\|_0)+\|c\|_{L^1}\Phi_p(\|x'\|_0)).
 \end{align*}
By primary inequality
\[
|a_1+a_2+\dots+a_n|^r\le C_r(|a_1|^r+\dots+|a_n|^r),\quad
C_r=\begin{cases}
 1,&0<r\le 1, \\
 n^{r-1},&r>1,\end{cases}
\]
we have
  \begin{align*}
\|Tx\|_0
&\leq 3^{q-1}\Delta_1\left[\Phi_p^{-1}(\|a\|_{L^1})+\Phi_p^{-1}(\|b\|_{L^1})\|x\|_0+\Phi_p^{-1}(\|c\|_{L^1})\|x'\|_0\right]\\
&\leq 3^{q-1}\Delta_1\left[\Phi_p^{-1}(\|a\|_{L^1})+(\Phi_p^{-1}(\|b\|_{L^1})+\Phi_p^{-1}(\|c\|_{L^1}))M\right]\\
&< M=\|x\|,
\end{align*}
and
\begin{align*}
\|(Tx)'\|_0
&\leq  3^{q-1}\Delta_2\left[\Phi_p^{-1}(\|a\|_{L^1})+\Phi_p^{-1}(\|b\|_{L^1})\|x\|_0+\Phi_p^{-1}(\|c\|_{L^1})\|x'\|_0\right]\\
&\leq 3^{q-1}\Delta_2\left[\Phi_p^{-1}(\|a\|_{L^1})+(\Phi_p^{-1}(\|b\|_{L^1})+\Phi_p^{-1}(\|c\|_{L^1}))M\right]\\
&< M=\|x\|.
\end{align*}
So $\|Tx\|< \|x\|$, i.e.  taking $p^*=0$ in
Lemma \ref{nonlinearalternative}, for any $x\in \partial\Omega$,
$x=\overline{\lambda}Tx$  ($0<\overline{\lambda}<1$) does not hold.
 Thus Lemma \ref{nonlinearalternative} implies that the operator
$T$ has at least one fixed point.
So  problem \eqref{eqm} has at least one positive
 solution. Besides, by $f(t, 0, 0)\neq 0$ for  $t\in[0, \infty)$,
 problem \eqref{eqm} has at least one nontrivial positive solution.
\end{proof}

\begin{corollary}\label{cor3423}
If $f(t, 0, 0)\not\equiv 0$ and there exists  $r>0$ such that
\begin{equation}\label{cor2}
\int_0^{\infty}f(t, x, y)dt
< \min\big\{\Phi_p\big(\frac{r}{\Delta_1}\big),
 \Phi_p\big(\frac{r}{\Delta_2}\big)\big\},
\end{equation}
where $x\in [0, r], y\in [-r, r]$.
 Then  \eqref{eqm} has at least one nontrivial positive solution.
\end{corollary}

\begin{proof}
From Lemma \ref{le2.4}, $T: P\to P$ is a
completely continuous operator.
 Now we define $\Omega=\{x\in P: \|x\|< r\}$. For any
$x\in \partial\Omega$,
$\|x\|=r$. So $\|x\|_0\le r, \|x'\|_0\le r$.
By assumption of  theorem and Lemma \ref{lm3.1},
 \begin{align*}
|Tx(t)|&=\frac{1}{1-\sum_{i=1}^{m}\alpha_i}\sum_{i=1}^{m}\alpha_i
\int_0^{\xi_i}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r),
x'(r))dr\Big)\Big]ds\\
&\quad +\int_0^{\infty}\Phi_p^{-1}\Big[\frac{1}{\rho(s)}\Big(
\rho(0)A_x-\int_0^{s}f(r, x(r), x'(r))dr\Big)\Big]ds
\\
&\leq \Delta_1\Phi_p^{-1}\Big(\int_0^{\infty}f(r, x(r),
x'(r))dr\Big)\\
&< \Delta_1\Phi_p^{-1}\Big(\min\Big\{\Phi_p\big(\frac{r}{\Delta_1}\big),
\Phi_p\big(\frac{r}{\Delta_2}\big)\Big\}\Big)\\
&\leq  r=\|x\|.
\end{align*}
\begin{align*}
|(Tx)'(t)|
&=\Big|\Phi_p^{-1}\Big[\frac{1}{\rho(t)}\Big(\rho(0)A_x-\int_0^{t}f(r,
x(r), x'(r))dr\Big)\Big]\Big|\\
&\leq \Delta_2\Phi_p^{-1}\Big(\int_0^{\infty}f(r, x(r),
x'(r))dr\Big)\\
&< \Delta_2\Phi_p^{-1}\Big(\min\Big\{\Phi_p\big(\frac{r}{\Delta_1}\big),
\Phi_p\big(\frac{r}{\Delta_2}\big)\Big\}\Big)\\
&\leq  r=\|x\|.
 \end{align*}
So $\|Tx\|< \|x\|$. Similar to the process in Theorem \ref{thm1},
the result follows.
\end{proof}

\begin{corollary} \label{coro3.3}
If $f(t, 0, 0)\not\equiv 0$ and
\begin{equation}\label{cor1cond2}
\lim_{d\to 0}\frac{\max_{x\in [0, d], y\in
[-d, d]}\int_0^{\infty}f(t, x, y)dt}{d^{p-1}}=0,
\end{equation}
then \eqref{eqm} has at least one nontrivial positive solution.
\end{corollary}

\begin{proof}
Let $\varepsilon^*=\min\{\Phi_p(\frac{1}{\Delta_1}),
\Phi_p(\frac{1}{\Delta_2})\}$.
By \eqref{cor1cond2}, there exists $r>0$, such that
\[
\max_{x\in [0, d], y\in [-d, d]}\int_0^{\infty}f(t, x, y)dt
\le \varepsilon^* d^{p-1}
=\min\big\{\Phi_p\big(\frac{d}{\Delta_1}\big),
\Phi_p\big(\frac{d}{\Delta_2}\big)\big\},\quad \forall
\, d\leq r,
\]
which implies \eqref{cor2}. By Corollary \ref{cor3423}, BVP
\eqref{eqm} has at  least one nontrivial
 positive solution.
\end{proof}

\section{Uniqueness of positive solutions}

  In this section, we  establish the uniqueness of
positive solutions for the problem
\begin{equation}\label{4.1}
\begin{gathered}
(\rho(t)\Phi_p(x'(t)))'+f(t, x(t))=0,\quad t\in I,\\
x(0)=\sum_{i=1}^{m}\alpha_i x(\xi_i), \quad
\lim_{t\to\infty}x(t)=0.
\end{gathered}
\end{equation}

\begin{lemma}\label{lem4.1}
Suppose that $f(t, x)$ is non-increasing in $x$ for all $t\in I$.
Then \eqref{4.1} has at most one positive solution.
\end{lemma}

\begin{proof}
Assume to the contrary, that \eqref{4.1} has two
positive solutions $x_1, x_2$. Let $y=x_2-x_1$.
Since $x_1, x_2$ are two positive solutions of \eqref{4.1},
\begin{gather*}
(\rho(t)\Phi_p(x_{2}'(t)))'-(\rho(t)\Phi_p(x_{1}'(t)))'=f(t,
x_1(t))-f(t, x_2(t)),\quad t\in I,\\
y(0)=\sum_{i=1}^{m}\alpha_i y(\xi_i), \quad
\lim_{t\to\infty}y(t)=0.
\end{gather*}
Let
$z(t)=\rho(t)\big(\Phi_p(x_{2}'(t))-\Phi_p(x_{1}'(t))\big)$.
Now we will complete the proof in three cases.

\noindent\textbf{Case 1.}
If $y(t)\ge0$, $y(t)\not \equiv 0$,  $t\in I$. Since
$f(t,x)$ is non-increasing in $x$, $z'(t)\ge0$, $t\in I$.
We claim that there
exists a unique $\eta\in I$ satisfying $z(\eta)=0$. If not, we get a
contradiction by the following  two cases.
\begin{itemize}
\item[(i)] $z(t)>0$, $t\in I$. Thus
$\Phi_p(x_{2}'(t))>\Phi_p(x_{1}'(t))$, $t\in I$; i.e.,
$y'(t)=(x_2-x_1)'(t)>0$, $t\in I$. So
$\lim_{t\to +\infty}y(t)>0$, a contradiction.

\item[(ii)] $z(t)<0$, $t\in I$. Thus
$\Phi_p(x_{2}'(t))<\Phi_p(x_{1}'(t))$, $t\in I$; i.e.,
$y'(t)=(x_2-x_1)'(t)<0$, $t\in I$. So
\begin{gather*}
y(0)=\sum_{i=1}^{m}\alpha_i y(\xi_i)\le \sum_{i=1}^{m}\alpha_i y(0)<y(0),
\quad \text{if }\sum_{i=1}^{m}\alpha_i\in (0, 1),\\
\lim_{t\to\infty}y(t)<y(0)=0,\quad \text{if }
\sum_{i=1}^{m}\alpha_i=0,
\end{gather*}
a contradiction. Our claim is proved.
\end{itemize}
So $z(t)<0$ for $t\in [0, \eta)$ and $z(t)>0$ for
$t\in (\eta, \infty)$. Thus
\begin{equation}\label{11}
\text{$y'(t)<0$ for $t\in [0, \eta)$ and
$y'(t)>0$ for $t\in (\eta, \infty)$.}
\end{equation}
If $\sum_{i=1}^{m}\alpha_i\in (0, 1)$,
we have by the first boundary condition,
\[
y(0)=\sum_{i=1}^{m}\alpha_i
y(\xi_i)\le\sum_{i=1}^{m}\alpha_iy(\xi_j)<y(\xi_j),
\]
where
$y(\xi_j)=\max\{y(\xi_i): i=1, 2, \dots, m\}$. So $\xi_j>\eta$. By
\eqref{11}, we have
\[
\lim_{t\to+\infty}y(t)\ge y(\xi_j)>y(0)\ge0,
\]
which contradicts the second boundary condition.

If $\sum_{i=1}^{m}\alpha_i=0$, we have
$0=\lim_{t\to\infty}y(t)>y(0)$, which  contradicts
the first boundary condition.


\noindent\textbf{Case 2.}
There exists $0<a<b$, $b\in (0, \infty]$ satisfying
$y(t)>0$ for $t\in (a, b)$, $y(a)=y(b)=0$,
$y'(a)\ge0$.
By the definition of $z(t)$, we have $z'(t)>0$, $t\in (a, b)$ and
$z(a)\ge0$. So $z(t)>0$, $t\in (a, b)$; i.e., $y'(t)>0$, $t\in (a, b)$.
By $y(a)=0$, we have $y(b)>0$, a contradiction.


\noindent\textbf{Case 3.} There exists $b\in (0, \infty)$ satisfying $y(t)>0$,
$t\in [0, b)$, $y(b)=0$, $y'(b)\le 0$.
By the definition of $z(t)$, we have $z'(t)>0$, $t\in [0, b)$ and
$z(b)\le0$. So $z(t)<0$, $t\in [0, b)$. Then $y'(t)=(x_2-x_1)'(t)<0$,
$t\in [0, b)$.

If $\sum_{i=1}^{m}\alpha_i\in (0, 1)$, we have by the first
boundary condition,
\[
0<y(0)=\sum_{i=1}^{m}\alpha_i y(\xi_i)
 <\sum_{i=1}^{m}\alpha_i y(\xi_j)<y(\xi_j),
\]
where $y(\xi_j)=\max\{y(\xi_i): i=1, 2, \dots, m\}$.
So $\xi_j>b$ and
$y(\xi_j)> y(0)>0$. So there exist $c, d$ satisfying
$b<c<\xi_j<d<\infty$ such that $y(t)>0$ for $t\in (c, d)$,
$y(c)=y(d)=0, y'(c)>0$. By Case 2, there is a contradiction.

If $\sum_{i=1}^{m}\alpha_i=0$, then $y(0)=0$, which
contradicts $y(t)>0, t\in [0, b)$.
\end{proof}

By Corollary \ref{cor3423} and Lemma \ref{lem4.1}, we have the
following result.

\begin{theorem}\label{thm4.1}
Suppose that $f(t, x)$ is nonincreasing in $x$ for all $t\in I$.
Also assume that there exists $r>0$ such that
\[
\int_0^{\infty}f(t, 0)dt<\Phi_p\big(\frac{r}{\Delta_1}\big).
\]
Then problem \eqref{4.1} has a unique positive solution.
\end{theorem}

\begin{thebibliography}{00}

\bibitem{agarwalzhuanzhuinfinite}R. P. Agarwal, D. O'Regan;
\emph{Infinite Interval Problems for
Differential, Difference and Integral Equations}, Kluwer Academic
Publishes, Dordrecht/Boston/London, 2001.


\bibitem{baichuanzhi}C. Z. Bai, J. X. Fang;
\emph{On positive solutions of boundary-value problems for second-order
functional differential equations on infinite intervals},
J. Math. Anal. Appl. 282 (2003) 711-731.

\bibitem{corduneanu} C. Corduneanu;
\emph{Integral Equations and Stability of Feedback Systems}, Academic
Press, New York, 1973.

\bibitem{duzengji}Z. J. Du, C. Y. Xue and W. G. Ge;
\emph{Multiple solutions for three-point boundary-value problem with
nonlinear terms depending on the first order derivative}, Arch. Math.
84 (2005) 341-349.

\bibitem{hege}X. He and W. Ge;
\emph{Triple solutions for second order three-point
boundary-value problems}. J. Math. Anal. Appl. 268 (2002) 256-265.

\bibitem{duodian}V. A. Il'in, E. I. Moiseev;
\emph{Nonlocal boundary-value problem of the
second kind for a Sturm¨CLiouville operator}, Differential Equations
23 (1987) 979-987.

\bibitem{madexiang} D. X. Ma, Z. J. Du, W. G. Ge;
\emph{Existence and iteration of monotone positive solutions
for multipoint boundary-value problem with p-Laplacian operator},
Comput. Math. Appl. 50 (2005) 729-739.

\bibitem{maruyun} R.Y. Ma and H. Y. Wang;
\emph{Positive solutions of nonlinear three-point boundary-value problems},
 J. Math. Anal. Appl. 279 (2003)
216-227.

\bibitem{meehan}M. Meehan, D. O'Regan;
\emph{Existence theory for nonlinear Fredholm and Volterra integral
equations on half-open intervals},  Nonlinear Anal. 35 (1999) 355-387.

\bibitem{tian1}Y. Tian, W. G. Ge, W. R. Shan;
\emph{Positive solutions for three-point boundary-value problem
on the half-line}, Comput. Math. Appl.  53(7), (2007) 1029-1039.


\bibitem{yanbaoqiang} B. Q. Yan;
\emph{Multiple unbounded solutions of boundary-value problems
for second-order differential equaitons on the half-line},
Nonlinear Anal. 51 (2002) 1031-1044.

\bibitem{yan}B. Q. Yan, Y. S. Liu;
\emph{Unbounded solutions of the singular boundary-value problems for
second order differential equations on the half-line}, Appl. Math.
Comput.  147 (2004) 629-644.

\bibitem{zima} M. Zima;
\emph{On positive solution of boundary-value problems on the half-line},
J. Math. Anal. Appl. 259(2001) 127-136.

\end{thebibliography}

\end{document}