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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 13, 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-2009/13\hfil Second-order m-point BVPs at resonance]
{Solvability for second-order m-point \\ boundary value problems
at resonance \\ on the half-line}

\author[Y. Liu, D. Li, M. Fang \hfil EJDE-2009/13\hfilneg]
{Yang Liu, Dong Li, Ming Fang} 

\address{Yang Liu \newline
  Department of  Mathematics \\
  Hefei Teachers College\\
  Hefei, Anhui 230061, China}
\email{liuyang19830206@yahoo.com.cn}

\address{Dong Li \newline
Department of  Mathematics \\
 Jiamusi University \\
  Jiamusi, Heilongjiang 154007, China}
\email{ld09281117@sohu.com}

\address{Ming Fang \newline
Department of Mathematics\\
  Yanbian University \\
  Yanji, Jilin 133000, China}
\email{fangming@ybu.edu.cn}

\thanks{Submitted July 8, 2008. Published January 12, 2009.}
\subjclass[2000]{34B15}
\keywords{m-point boundary value problem;
resonance; half-line; \hfill\break\indent coincidence degree theory}

\begin{abstract}
 In this article, we investigate the  existence of positive
 solutions for second-order m-point boundary-value problems
 at resonance on the half-line
 \begin{gather*}
 (q(t)x'(t))'=f(t,x(t),x'(t)),\quad \text{a.e. in }(0,\infty), \\
 x(0)=\sum_{i=1}^{m-2}\alpha_ix(\xi_i),\quad
 \lim_{t\to \infty}q(t)x'(t)=0.
 \end{gather*}
 Some existence results are obtained by using the Mawhin's
 coincidence theory.
\end{abstract}

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

\section{Introduction}

In this article, we study the existence of positive solutions for
the second-order m-point boundary-value problems at resonance on
the half-line
\begin{equation}
(q(t)x'(t))'=f(t,x(t),x'(t)),\quad a.e.\hspace{1mm}
\rm{in}\hspace{1mm}(0,\infty), \label{e1.1}
\end{equation}
\begin{equation}
x(0)=\sum_{i=1}^{m-2}\alpha_ix(\xi_i),\quad \lim_{t\to
\infty}q(t)x'(t)=0,
 \label{e1.2}
\end{equation}
 where $f:[0,\infty)\times \mathbb{R}^2\to \mathbb{R}$ is a
Carath\'{e}odory function, $\alpha_i\in \mathbb{R}$
$(1\le i\le m-2)$, $0<\xi_1<\xi_2<\cdots<\xi_{m-2}<1$,
$q\in C[0,\infty)\cap C^1(0,\infty)$ with $q>0$ on $[0,\infty)$
 and $\frac{1}{q}\in L_1[0,\infty)$.

In recent years, many authors have studied the existence of
positive solutions for some boundary value problems on the half-line
(see \cite{k2,l1,o1,y1,y2,z1}) or at resonance (see
\cite{b1,b2,d1,k1,l3,l4}). However, to the best of our knowledge,
only one paper \cite{l2} studied the existence and uniqueness
positive solutions for second-order three-point boundary value
problems at resonance on the half-line. There is little research
concerning (\ref{e1.1})-(\ref{e1.2}), so it is worthwhile to
investigate the problem.

Inspired by \cite{b1,d1,k1}, the purpose of our paper is to discuss
the existence of positive solutions for the second-order m-point
boundary value problem at resonance on the half-line. Our method is
based on the coincidence degree theory of Mawhin.

The remaining part of this paper is organized as follows. In section
2, we present some preliminaries and lemmas. Section 3 is devoted to
proving the existence of positive solutions for
(\ref{e1.1})-(\ref{e1.2}).

\section{Preliminaries and lemmas}

Now, we briefly recall some notation and an abstract existence
result.

Let $X$, $Z$ be normed spaces, $L: \mathop{\rm dom}L\subset X \to Z$ be a
Fredholm operator of index zero, and $P: X\to X$, $Q: Z\to Z$ be
continuous projectors such that $\mathop{\rm Im} P=\ker L$, $\ker Q=\mathop{\rm Im} L$ and $X=\ker L\oplus\ker P$, $Z=\mathop{\rm Im} L\oplus\mathop{\rm Im} Q$,
It follows that $L|_{\mathop{\rm dom} L\cap \ker P}: {\rm dom } L\cap \ker
P\to \mathop{\rm Im}L$ is invertible. We denote the inverse of the mapping
by $K_P: \mathop{\rm Im}L \to {\rm dom } L\cap \ker P$. The generalized
inverse of $L$ denoted by $K_{P,Q}: Z \to {\rm dom } L\cap \ker P$
is defined by $K_{P,Q}=K_p(I-Q)$.

\begin{definition}\label{def2.1}
{\rm Let $L: \mathop{\rm dom}L\subset X\to Z$ be a Fredholm mapping, $E$ be
a metric space, and $N:E\to Z$ be a mapping. We say that $N$ is
$L$-compact on $E$ if $QN:E\to Z$ and $K_{P,Q}N:E\to X$ are compact
on $E$. In addition, we say that $N$ is $L$-completely continuous if
it is $L$-compact on every bounded $E\subset X$.}
\end{definition}

\begin{definition}\label{def2.2}
\rm We say that the map $f:[0,\infty)\times \mathbb{R}^n\to
\mathbb{R}$, $(t,x)\to f(t,z)$ is $L_1[0,\infty)$-Carath\'{e}odory,
if the following conditions are satisfied
\begin{itemize}
\item[(i)] for each $z\in \mathbb{R}^n$, the mapping
$t\to f(t,z)$ is Lebesgue measurable;

\item[(ii)] for a.e. $t\in [0,\infty)$, the mapping $z\to
f(t,z)$ is continuous on $\mathbb{R}^n$;

\item[(iii)] for each $r>0$, there exists $\varphi_r\in
L_1[0,\infty)$ such that, for a.e. $t\in[0,\infty)$ and every $z$
such that $|z|\le r$, we have $|f(t,z)|\le \varphi_r(t)$.
\end{itemize}
\end{definition}

\begin{lemma}[\cite{a1}]\label{lem2.3}
 Let $X$ be the space of all bounded continuous
vector-value functions on $[0,\infty)$ and $M\subset X$. Then $M$ is
relatively compact in $X$ if the following conditions hold:
\begin{itemize}
\item[(i)] $M$ is bounded in $X$:
\item[(ii)] the functions from $M$ are equicontinuous on any
compact interval of $[0,\infty)$;
\item[(iii)] the functions from $M$ are equiconvergent, that is,
given $\epsilon>0$, there exists a $T=T(\epsilon)>0$ such that
$|\phi(t)-\phi(\infty)|<\epsilon$, for all $t>T$ and all
$\phi\in S$.
\end{itemize}
\end{lemma}

\begin{lemma}[\cite{m1}]\label{lem2.4}
 Let $\Omega\subset X$ be open and bounded, $L$ be a
Fredholm mapping of index zero and $N$ be L-compact on
$\overline\Omega$. Assume that the following conditions are
satisfied:
\begin{itemize}
\item[(1)] $Lx\neq \lambda Nx$ for every $(x,\lambda)\in [(\mathop{\rm dom} L\setminus \ker L)\cap
\partial \Omega]\times (0,1)$;

\item[(2)] $Nx \not\in\mathop{\rm Im} L$ for every $x\in \ker
L\cap\partial\Omega$;

\item[(3)] $\deg (JQN|_{\partial\Omega\cap \ker L},\Omega\cap \ker
L,0)\neq 0$, with $Q:Z\to Z$ is a continuous projection such that
$\mathop{\rm Im} L=\ker Q$ and $J: \mathop{\rm Im} Q\to \ker L$ is an
isomorphism.
\end{itemize}
Then the equation $Lx=Nx$ has at least one solution in $\mathop{\rm dom}
L\cap \overline \Omega$.
\end{lemma}

Let $AC[0,\infty)$ denote the space of absolutely continuous
functions on the interval $[0,\infty)$. In this paper, the following
space $X$ will be basic space to study (\ref{e1.1})-(\ref{e1.2}),
which is denoted by
\begin{align*}
X&=\{ x\in C^1[0,\infty), x,qx'\in AC[0,\infty) \lim_{t\to \infty}x(t)\\
 &\quad \text{and }  \lim_{t\to \infty}x'(t)
 \text{exist}, \;(qx')'\in L_1[0,\infty)\}
\end{align*}
endowed with the norm $\|x\|=\max\{\|x\|_\infty,\|x'\|_\infty\}$,
where $\|x\|_{\infty}=\sup_{t\in[0,\infty)}|x(t)|$.

Let $Z=L_1[0,\infty)$, and denote the norm in $L_1[0,\infty)$ by
$\|\cdot\|_1$.

Define $L$ to be the linear operator from $L\subset X$ to $Z$ with
$$
\mathop{\rm dom}L=\{x\in X: x(0)=\sum_{i=1}^{m-2}\alpha_ix(\xi_i),
\lim_{t\to \infty}q(t)x'(t)=0\}
$$
and $Lx(t)=(q(t)x'(t))'$, $x\in \mathop{\rm dom}L$, $t\in[0,\infty)$. We
define $N:X\to Z$ by setting
$$
Nx(t)=f(t,x(t),x'(t)), \quad t\in[0,\infty),
$$
then  (\ref{e1.1})-(\ref{e1.2}) can be written
$$
Lx=Nx
$$

\begin{lemma}\label{lem2.5}
 If $\sum_{i=1}^{m-2}\alpha_i=1$ and
$\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}\frac{e^{-s}}{q(s)}ds\neq0$,
then
\begin{itemize}
\item[(i)] $\ker L=\{x\in \mathop{\rm dom}L:x(t)=c,
c\in\mathbb{R},t\in[0,\infty)\}$;

\item[(ii)] $\mathop{\rm Im} L=\{y\in
Z:\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}\frac{1}{q(s)}\int_s^{\infty}y(\tau)d\tau
ds=0\}$;

\item[(iii)] $L:\mathop{\rm dom} L\subset X\to X$ is a Fredholm
operator of index zero. Furthermore, the linear continuous projector
operator $Q:Z\to Z$ can be defined
$$(Qy)(t)=h(t)\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}\frac{1}{q(s)}
\int_s^{\infty}y(\tau)d\tau ds,\quad t\in [0,\infty),$$ where
\[
h(t)=\frac{e^{-t}}{\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}
\frac{e^{-s}}{q(s)}ds},
\quad t\in [0,\infty).
\]

\item[(iv)] The generalized inverse $K_{P}:\mathop{\rm Im}L\to
\mathop{\rm dom}L\cap \ker P$ of $L$ can be written by
$$
K_Py(t)=-\int_0^t\frac{1}{q(s)}\int_s^{\infty}y(\tau)d\tau
ds.
$$

\item[(v)] $\|K_Py\|\le
\max\{\|q^{-1}\|_\infty,\|q^{-1}\|_1\}\|y\|_1$, for all $y\in {\rm
Im L}$.
\end{itemize}
\end{lemma}

\begin{proof}
By direct calculations, we easily know that
(i) and (ii) hold.
(iii) For any $y\in Z$, take the prosector
$$
(Qy)(t)=h(t)\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}\frac{1}{q(s)}
\int_s^{\infty}y(\tau)d\tau ds,\quad t\in [0,\infty).
$$
Let
$y_1=y-Qy$, by direct calculations, we have
\begin{align*}
&\sum_{i=1}^{m-2}\int_0^{\xi_i}\frac{1}{q(s)}\int_s^{\infty}y_1(\tau)d\tau
ds\\
&=\sum_{i=1}^{m-2}\int_0^{\xi_i}\frac{1}{q(s)}\int_s^{\infty}y(\tau)d\tau
ds\Big(1-\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}\frac{1}{q(s)}
\int_s^{\infty}h(\tau)d\tau ds\Big)
=0.
\end{align*}
So $y_1\in \mathop{\rm Im}L$. Hence, $Z=\mathop{\rm Im}L+\mathop{\rm Im}Q$, since $\mathop{\rm Im}L\cap \mathop{\rm Im}Q=\{0\}$, we obtain
$$
Z=\mathop{\rm Im}L\oplus \mathop{\rm Im}Q.$$
Thus, ${\rm dim} \ker L={\rm dim }\mathop{\rm Im}Q=1$. \\
Hence, $L$ is a Fredholm operator of index zero.

${\rm (iv)}$ Let $P:Z\to Z$ be defined by
$$
Px(t)=x(0),\quad t\in[0,\infty).
$$
Then the generalized inverse
$K_P:\mathop{\rm Im}L\to \mathop{\rm dom}L\cap \ker P$ of $L$ can be written as
$$
K_Py(t)=-\int_0^t\frac{1}{q(s)}\int_s^{\infty}y(\tau)d\tau
ds.
$$
In fact, for any $y\in {\rm Im }L$, we have
$$
LK_Py(t)=(q(t)K_Py'(t))'=y(t).
$$
and for $x\in \mathop{\rm dom} L\cap \ker P$, one has
\begin{align*}
K_PLx(t)=K_P(q(t)x'(t))'&=-\int_0^t\frac{1}{q(s)}\int_s^{\infty}
(q(\tau)x'(\tau))'d\tau ds\\
&=-\int_0^t\frac{1}{q(s)}\left(\lim_{\sigma\to
\infty}q(\sigma)x'(\sigma)-q(s)x'(s)\right)ds\\
&=\int_0^tx'(s)ds=x(t)-x(0),
\end{align*}
in view of $x(0)=0$ (since $x\in \ker P$), thus,
$$
(K_PL)x(t)=x(t),\quad t\in[0,\infty).
$$
Hence, $K_P=(L|_{\mathop{\rm dom}L\cap \ker P})^{-1}$.

${\rm (v)}$ From the definition of $K_P$, we have
$$
\|K_Py\|_{\infty}=\sup_{t\in[0,\infty)}|K_Py|\le
\sup_{t\in[0,\infty)}\int_0^t\frac{1}{q(s)}\int_s^{\infty}|y(\tau)|d\tau
ds\le \|q^{-1}\|_1\|y\|_1,
$$
and
$$
\|(K_Py)'\|_{\infty}=\sup_{t\in[0,\infty)}|(K_Py)'|\le
\sup_{t\in[0,\infty)}\frac{1}{q(t)}\int_t^{\infty}|y(s)| ds\le
\|q^{-1}\|_{\infty}\|y\|_1.
$$
Hence,
$$
\|K_Py\|\le\max\{\|q^{-1}\|_1,\|q^{-1}\|_{\infty}\}\|y\|_1.
$$
\end{proof}

\begin{lemma}\label{lem2.6}
If $f$ is a Carath\'{e}odory function and
$\sum_{i=1}^{m-2}|\alpha_i|\int_0^{\xi_i}\frac{1}{q(s)}ds<\infty$,
then $N$ is $L$-compact.
\end{lemma}

\begin{proof}
Let $M\subset X$ be bounded with $r=\sup \{\|x\|:x\in M\}$ and consider
$K_{P,Q}N(M).$ By $f:[0,\infty)\times \mathbb{R}^2\to \mathbb{R}$
satisfies the Carath\'{e}odory
conditions with respect to $L_1[0,\infty)$, there exists a Lebesgue
integrable function $\varphi_r$ such that
$$|Nx(t)|=|f(t,x(t),x'(t))|\le \varphi_r(t)\quad {\rm a.e.}\quad {\rm
in}\hspace{1mm} (0,\infty).$$ Then for all $x\in M$, we have
\begin{align*}
\|QNx\|_1&\le\int_0^{\infty}|QNx(s)|ds\\
&=\int_0^{\infty}\Big|h(s)\sum_{i=1}^{m-2}\alpha_i
\int_0^{\xi_i}\frac{1}{q(\varsigma)}\int_{\varsigma}^{\infty}
f(\tau,x(\tau),x'(\tau))d\tau d\varsigma\Big| ds\\
&\le \int_0^{\infty}|h(s)|\sum_{i=1}^{m-2}|\alpha_i|
\int_0^{\xi_1}\frac{1}{q(\varsigma)}\int_0^{\infty}\varphi_{r}(\tau)d\tau
d\varsigma ds\\
&\le
\|h\|_1\|\varphi_r\|\sum_{i=1}^{m-2}|\alpha_i|\int_0^{\xi_i}\frac{1}{q(\varsigma)}d\varsigma
<\infty.
\end{align*}
Thus,
\begin{align*}
&\|K_{P,Q}Nx\|_{\infty}\\
&=\Big|\sup_{t\in[0,\infty)}\int_0^t\frac{1}{q(s)}\int_s^{\infty}
\Big(f(\tau,x(\tau),x'(\tau))\\
&\quad -h(\tau)\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}\frac{1}{q(\varsigma)}
\int_{\varsigma}^{\infty}f(\zeta,x(\zeta),x'(\zeta))d\zeta
d\varsigma\Big) d\tau ds\Big|\\
&\le\sup_{t\in[0,\infty)}\int_0^t\frac{1}{q(s)}\int_s^{\infty}\Big|
f(\tau,x(\tau),x'(\tau))\\
&\quad -h(\tau)\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}\frac{1}{q(\varsigma)}
\int_{\varsigma}^{\infty}f(\zeta,x(\zeta),x'(\zeta))d\zeta
d\varsigma \Big|d\tau ds\\
&\le\int_0^{\infty}\frac{1}{q(s)}
\int_0^{\infty}\Big(\varphi_r(\tau)+|h(\tau)|\sum_{i=1}^{m-2}|\alpha_i|
\int_0^{\xi_i}\frac{1}{q(\varsigma)}\int_{0}^{\infty}\varphi_r(\zeta)d\zeta
d\varsigma\Big) d\tau ds\\
&\le\|\varphi_r\|_1\|q^{-1}\|_1\Big(1+\|h\|_1\sum_{i=1}^{m-2}|\alpha_i|
\int_0^{\xi_i}\frac{1}{q(\varsigma)}d\varsigma\Big)<\infty,
\end{align*}
and
\begin{align*}
&\|(K_{P,Q}Nx)'\|_{\infty}\\
&=\sup_{t\in[0,\infty)}\Big|\frac{1}{q(t)}
\int_t^{\infty}\Big(f(s,x(s),x'(s))\\
&\quad -h(s)\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}
\frac{1}{q(\varsigma)}\int_{\varsigma}^{\infty}f(\tau,x(\tau),x'(\tau))d\tau
d\varsigma\Big)ds\Big|\\
&\le\sup_{t\in[0,\infty)}\frac{1}{q(t)}
\int_0^{\infty}\Big(\varphi_r(s)+|h(s)|
\sum_{i=1}^{m-2}|\alpha_i|\int_0^{\xi_i}
\frac{1}{q(\varsigma)}\int_{0}^{\infty}\varphi_r(\tau)d\tau
d\varsigma\Big)ds\\
&\le\|q^{-1}\|_{\infty}\|\varphi_r\|_1\Big(1+\|h\|_1\sum_{i=1}^{m-2}
|\alpha_i|\int_0^{\xi_i}\frac{1}{q(\varsigma)}d\varsigma\Big)<\infty.
\end{align*}
It follows that $K_{P,Q}N(M)$ is uniformly bounded in $X$.

Let $x\in M$ and $t_1$, $t_2\in [0,T]$ with $T\in (0,\infty)$, we
have
\begin{align*}
&|K_{P,Q}Nx(t_2)-K_{P,Q}Nx(t_1)|\\
&=\Big|\int_{t_1}^{t_2}\frac{1}{q(s)}
\int_s^{\infty}\Big(f(\tau,x(\tau),x'(\tau))\\
&\quad -h(\tau)\sum_{i=1}^{m-2}\alpha_i
\int_0^{\xi_i}\frac{1}{q(\varsigma)}\int_{\varsigma}^{\infty}f(\zeta,x(\zeta),x'(\zeta))
d\zeta d\varsigma \Big)d\tau ds\Big|\\
&\le \int_{t_1}^{t_2}\frac{1}{q(s)}
\int_0^{\infty}\Big(\varphi_r(\tau)+|h(\tau)|\sum_{i=1}^{m-2}|\alpha_i|
\int_0^{\xi_i}\frac{1}{q(\varsigma)}\int_{0}^{\infty}\varphi_r(\zeta)
d\zeta d\varsigma \Big)d\tau ds\\
&\le \int_{t_1}^{t_2}\frac{1}{q(s)}
\|\varphi_r\|_1\Big(1+\|h\|_1\sum_{i=1}^{m-2}|\alpha_i|
\int_0^{\xi_i}\frac{1}{q(\varsigma)}d\varsigma  \Big)ds\to 0,\quad
\text{as }t_1\to t_2,
\end{align*}
and
\begin{align*}
&|(K_{P,Q}Nx)'(t_2)-(K_{P,Q}Nx)'(t_1)|\\
&=\Big|\frac{1}{q(t_2)}\int_{t_2}^{\infty}
 \Big(f(s,x(s),x'(s))-h(s)\sum_{i=1}^{m-2}
\alpha_i\int_0^{\xi_i}\frac{1}{q(\varsigma)}\int_{\varsigma}^{\infty}
\! f(\tau,x(\tau),x'(\tau))d\tau d\varsigma\Big)ds \\
&\quad-\frac{1}{q(t_1)}\int_{t_1}^{\infty}\Big(f(s,x(s),x'(s))\\
&\quad -h(s)\sum_{i=1}^{m-2}
\alpha_i\int_0^{\xi_i}\frac{1}{q(\varsigma)}\int_{\varsigma}^{\infty}
f(\tau,x(\tau),x'(\tau))d\tau d\varsigma\Big)ds\Big|\\
&\le\Big|\frac{1}{q(t_2)}-\frac{1}{q(t_1)}\Big|\int_{t_2}^{\infty}
\Big(|f(s,x(s),x'(s))|\\
&\quad +|h(s)|\sum_{i=1}^{m-2}|\alpha_i|\int_0^{\xi_i}\frac{1}{q(\varsigma)}
\int_{\varsigma}^{\infty}|f(\tau,x(\tau),x'(\tau))|d\tau
d\varsigma\Big)ds\\
&\quad+\frac{1}{q(t_1)}\int_{t_1}^{t_2}\Big(|f(s,x(s),x'(s))|\\
&\quad +|h(s)|\sum_{i=1}^{m-2}
|\alpha_i|\int_0^{\xi_i}\frac{1}{q(\varsigma)}\int_{\varsigma}^{\infty}
|f(\tau,x(\tau),x'(\tau))|d\tau d\varsigma\Big)ds\\
&\le\|q^{-1}\|^2_{\infty}|q(t_1)-q(t_2)|\|\varphi_r\|_1\Big(
1+\|h\|_1\sum_{i=1}^{m-2}|\alpha_i|\int_0^{\xi_i}\frac{1}{q(\varsigma)}d\varsigma\Big)\\
&\quad+\|q^{-1}\|_{\infty}\int_{t_1}^{t_2}\Big(\varphi_r(s))+|h(s)|\sum_{i=1}^{m-2}
|\alpha_i|\int_0^{\xi_i}\frac{1}{q(\varsigma)}d\varsigma\|\varphi_r\|_1\Big)ds\to
0,\quad \text{as } t_1\to t_2.
\end{align*}
So $K_{P,Q}N(E)$ is equicontinuous on every compact subset of
$[0,\infty)$.

We introduce the following notation:
\begin{align*}
&K_{P,Q}Nx(\infty)=\lim_{t\to \infty}K_{P,Q}Nx(t)\\
&=\int_0^{\infty}\frac{1}{q(s)}\int_s^{\infty}
\Big(f(\tau,x(\tau),x'(\tau))\\
&\quad -h(\tau)\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}
\frac{1}{q(\varsigma)}\int_{\varsigma}^{\infty}f(\zeta,x(\zeta),x'(\zeta))
d\zeta d\varsigma  \Big)d\tau ds,
\end{align*}
and
\begin{align*}
&(K_{P,Q}Nx)'(\infty)=\lim_{t\to \infty}(K_{P,Q}Nx)'(t)\\
&=\lim_{t\to \infty}\frac{1}{q(t)}\int_t^{\infty}
\Big(f(s,x(s),x'(s))\\
&\quad -h(s)\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_I}
\frac{1}{q(\varsigma)}\int_{\varsigma}^{\infty}f(\tau,x(\tau),x'(\tau))
d\tau d\varsigma  \Big)ds=0.
\end{align*}
Thus,
\begin{align*}
&|K_{P,Q}Nx(t)-K_{P,Q}Nx(\infty)|\\
&=\Big|\int_t^{\infty}\frac{1}{q(s)}\int_s^{\infty}
\Big(f(\tau,x(\tau),x'(\tau))\\
&\quad -h(\tau)\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}
\frac{1}{q(\varsigma)}\int_{\varsigma}^{\infty}f(\zeta,x(\zeta),x'(\zeta))
d\zeta d\varsigma  \Big)d\tau ds\Big|\\
&\le\int_t^{\infty}\frac{1}{q(s)}\int_s^{\infty}
\Big(\varphi_r(\tau)+|h(\tau)|\sum_{i=1}^{m-2}|\alpha_i|\int_0^{\xi_i}
\frac{1}{q(\varsigma)}\int_{\varsigma}^{\infty}\varphi_r(\zeta)
d\zeta d\varsigma  \Big)d\tau ds\\
&\le\int_t^{\infty}\frac{1}{q(s)}\|\varphi\|_1\Big(1+\|h\|_1
\sum_{i=1}^{m-2}|\alpha_i|\int_0^{\xi_i}\frac{1}{q(\tau)}d\tau\Big)ds\to0,
\quad \text{uniformly as }t\to \infty,
\end{align*}
and
\begin{align*}
&|(K_{P,Q}Nx)'(t)-(K_{P,Q}Nx)'(\infty)|\\
&=\Big|\frac{1}{q(t)} \int_t^{\infty}
\Big(f(s,x(s),x'(s))-h(s)\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_I}
\frac{1}{q(\varsigma)}\int_{\varsigma}^{\infty}\! f(\tau,x(\tau),x'(\tau))
d\tau d\varsigma  \Big)ds\Big|\\
&\le\frac{1}{q(t)} \int_t^{\infty}
\Big(\varphi_r(s)+|h(s)|\sum_{i=1}^{m-2}|\alpha_i|\int_0^{\xi_I}
\frac{1}{q(\varsigma)} d\varsigma \|\varphi_r\|_1\Big)ds\to0,
\end{align*}
 uniformly as $t\to \infty$.
Therefore, $K_{P,Q}N(M)$ is equiconvergent. It follows from Lemma
\ref{lem2.3} that $K_{P,Q}N(M)$ is relatively compact for each
bounded $M\in X$. The continuity of $K_{P,Q}N(M)$ follows from the
Lebesgue Dominated Theorem. We can easily see that $QN$ is
continuous and $QN(M)$ is relatively compact. Thus, by Definition
\ref{def2.1}, we have that the mapping $N:X\to Z$ is $L$-completely
continuous.
\end{proof}

\section{Main results}

\begin{theorem}\label{thm3.1}
Let $f:[0,\infty)\times \mathbb{R}^2\to \mathbb{R}$ be a
Carath\'{e}odory function, in addition, assume that\\
${\rm(H_0)}$ $\sum_{i=1}^{m-2}\alpha_i=1$,
$\sum_{i=1}^{m-2}|\alpha_i|\int_0^{\xi_i}\frac{1}{q(s)}ds<\infty$
and
$\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}\frac{e^{-s}}{q(s)}ds\neq0$;
\begin{itemize}

\item[(H1)] There exists a constant $M>0$, such that for
all $x\in \mathop{\rm dom}L\setminus \ker L$ if $|x(t)|>M$, $t\in
[0,\infty)$, then
\begin{equation}\label{e3.1}
h(t)\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}\frac{1}{q(s)}\int_s^{
\infty}f(\tau,x(\tau),x'(\tau))d\tau ds\neq0
\end{equation}

\item[(H2)] There exist
$\beta,\gamma,\delta,\rho:[0,\infty)\to [0,\infty)$,
$\beta,\gamma,\delta,\rho\in L_1[0,\infty)$, and constant $\theta\in
[0,1)$, such that for all $(x_1,x_2)\in \mathbb{R}^2$, $t\in
[0,\infty)$ satisfying one of the following inequalities
\begin{gather}\label{e3.2}
|f(t,x_1,x_2)|\le
\beta(t)|x_1|+\gamma(t)|x'|+\delta(t)|x_2|^{\theta}+\rho(t),
\\ \label{e3.3}
|f(t,x_1,x_2)|\le
\beta(t)|x_1|+\gamma(t)|x'|+\delta(t)|x_1|^{\theta}+\rho(t),
\end{gather}

\item[(H3)] There exists a constant $N^*>0$, such that
for all $c\in \mathbb{R}$, if $|c|>N^*$, then, either
\begin{equation}\label{e3.4}
c\sum_{i=1}^{m-2}\alpha_i\int_{0}^{\xi_i}\frac{1}{q(s)}\int_s^{\infty}f(\tau,c,0)d\tau
ds<0,
\end{equation}
or
\begin{equation}\label{e3.5}
c\sum_{i=1}^{m-2}\alpha_i\int_{0}^{\xi_i}\frac{1}{q(s)}\int_s^{\infty}f(\tau,c,0)d\tau
ds>0.
\end{equation}
\end{itemize}
Then  \rm{(\ref{e1.1})-(\ref{e1.2})} has at least one solution if
$$
\max\{2\|q^{-1}\|_1,\|q^{-1}\|_1+\|q^{-1}\|_{\infty}\}(\|\beta\|_1
+\|\gamma\|_1)<1.
$$
\end{theorem}

\begin{proof}   Set
$$
\Omega_1=\{x\in \mathop{\rm dom}L\setminus \ker
L:Lx=\lambda Nx, \lambda\in [0,1]\}.
$$
For $x\in \Omega_1$, since $Lx=\lambda Nx$, thus, $\lambda\neq0$,
$Nx\in \mathop{\rm Im}L=\ker Q$,
hence,
$$
h(t)\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}f(\tau,x(\tau),x'(\tau))d\tau
ds=0.
$$
Thus, by (H1), there exists $t_0\in [0,\infty)$,
such that $|x(t_0)|\le M$. In view of
$$
|x(0)|=|x(t_0)-\int_0^{t_0}x'(s)ds|\le M+\|x'\|_1.
$$
In addition,
$$
x'(t)=-\frac{1}{q(t)}\int_t^{\infty}(q(s)x'(s))'ds
=-\int_t^{\infty}Lx(s)ds,
$$
which implies
$$
\|x'\|_{\infty}=\sup_{t\in[0,\infty)}\big|-\frac{1}{q(t)}
\int_t^{\infty}Lx(s)ds\big|
\le \|q^{-1}\|_{\infty}\|Lx\|_1
\le \|q^{-1}\|_{\infty}\|Nx\|_1,
$$
and
$$
\|x'\|_{1}=\int_0^{\infty}\Big|-\frac{1}{q(\tau)}
\int_{\tau}^{\infty}Lx(s)ds\Big|d\tau
\le \|q^{-1}\|_{1}\|Lx\|_1
\le \|q^{-1}\|_{1}\|Nx\|_1.
$$
Thus,
\begin{equation}
|x(0)|\le M+\|q^{-1}\|_1\|Nx\|_1. \label{e3.6}
\end{equation}
Again for all $x\in \Omega_1$, $(I-P)x\in\mathop{\rm dom}L\cap \ker P$,
$LPx=0$, thus, from Lemma \ref{lem2.4}, we get
\begin{equation}\label{e3.7}
\begin{aligned}
\|(I-P)x\|=\|K_{P}(I-P)x\|
&\le \max\{\|q^{-1}\|_1,\|q^{-1}\|_{\infty}\}\|L(I-P)x\|_1\\
&=\max\{\|q^{-1}\|_1,\|q^{-1}\|_{\infty}\}\|Lx\|_1\\
&\le\max\{\|q^{-1}\|_1,\|q^{-1}\|_{\infty}\}\|Nx\|_1.
\end{aligned}
\end{equation}
Hence, we have from (\ref{e3.1}) that
\begin{equation}\label{e3.8}
\begin{aligned}
\|x\|&\le\|Px\|+\|(I-P)x\|\\
&\le M+\|q^{-1}\|_1\|Nx\|_1+\max\{\|q^{-1}\|_1,\|q^{-1}\|_{\infty}\}
 \|Nx\|_1\\
&\le M+ \max\{2\|q^{-1}\|_1,\|q^{-1}\|_1+\|q^{-1}\|_{\infty}\}\|Nx\|_1.
\end{aligned}
\end{equation}
Let
$\Lambda=\max\{2\|q^{-1}\|_1,\|q^{-1}\|_1+\|q^{-1}\|_{\infty}\}$. If
(\ref{e3.2}) holds, then from (\ref{e3.8}), we get
\begin{equation}\label{e3.9}
\|x\|\le M+\Lambda\|Nx\|_1\le
M+\Lambda(\|\beta\|_1\|x\|_{\infty}+\|\gamma\|_1\|x'\|_{\infty}+
\|\delta\|_1\|x'\|^{\theta}_{\infty}+\|\rho\|_1).
\end{equation}
Thus, from $\|x\|_{\infty}\le \|x\|$ and (\ref{e3.9}), we have
\begin{equation}\label{e3.10}
\|x\|_{\infty}\le \frac{M+\Lambda(\|\beta\|_1\|x\|_{\infty}
+\|\gamma\|_1\|x'\|_{\infty}+\|\delta\|_1\|x'\|^{\theta}_{\infty}+\|\rho\|_1)}
{1-\Lambda\|\beta\|_1}.
\end{equation}
It follows from $\|x'\|_{\infty}\le \|x\|$, (\ref{e3.9}) and
(\ref{e3.10}) that
\begin{align*}
\|x'\|_{\infty}&\le
\Lambda\|\beta\|_1\|x\|_{\infty}+\Lambda\Big(\|\gamma\|_1\|x'\|_{\infty}+
\|\delta\|_1\|x'\|^{\theta}_{\infty}+\|\rho\|_1+\frac{M}{\Lambda}\Big)\\
&\le\frac{\Lambda\|\gamma\|_1}{1-\Lambda\|\beta\|_1}\|x'\|_{\infty}
+\frac{\Lambda\|\delta\|_1}{1-\Lambda\|\beta\|_1}\|x'\|^{\theta}_{\infty}
+\frac{\Lambda\|\rho\|_1+M}{1-\Lambda\|\beta\|_1}.
\end{align*}
So
\begin{equation}\label{e3.11}
\|x'\|_{\infty}\le\frac{\Lambda\|\delta\|_1}{1-\Lambda(\|\beta\|_1
+\|\gamma\|_1)}
\|x'\|^{\theta}_{\infty}+\frac{\Lambda\|\rho\|_1+M}
{1-\Lambda(\|\beta\|_1+\|\gamma\|_1)}.
\end{equation}
Since $\theta\in [0,1)$, by (\ref{e3.11}), there exists $M_1>0$,
such that
\begin{equation}\label{e3.12}
\|x'\|_{\infty}\le M_1.
\end{equation}
Similar, by (\ref{e3.10}) and (\ref{e3.12}), there exists $M_2>0$,
such that
\begin{equation}\label{e3.13}
\|x\|_{\infty}\le M_2.
\end{equation}
Hence,
$$
\|x\|=\max\{\|x\|_{\infty},\|x'\|_{\infty}\}\le
\max\{M_1,M_2\}.
$$
 Then $\Omega_1$ is bounded.

If (\ref{e3.3}) holds, similar to the above argument, we can prove
that $\Omega_1$ is bounded too.
Let
$$
\Omega_2=\{x\in \ker L:Nx\in \mathop{\rm Im} L\}.
$$
For $x\in \Omega_2$, then we have $x=c \in \mathbb{R}$, thus,
\begin{equation}\label{e3.14}
\sum_{i=1}^{m-2}\alpha_i\int_0^{\xi_i}\frac{1}{q(s)}
\int_s^{\infty}f(\tau,c,0) d\tau ds=0.
\end{equation}
Then, we have by (H3) and (\ref{e3.14}) that
$$
\|x\|=|c|\le N^*,
$$
which implies that $\Omega_2$ is bounded.
We define the isomorphism $J:\mathop{\rm Im}Q\to \ker L$ by
$$
J(ch(t))=c,\quad c\in \mathbb{R},\; t\in [0,\infty).
$$
If (\ref{e3.4}) holds, set
$$
\Omega_3=\{x\in \ker L:-\lambda x+(1-\lambda)JQNx=0, \;\lambda\in [0,1]\}.
$$
For every $c_0\in \Omega_3$, we obtain
$$
\lambda c_0=(1-\lambda)\sum_{i=1}^{m-2}\alpha_i
\int_0^{\xi_i}\frac{1}{q(s)}\int_s^{\infty}f(\tau,c_0,0)d\tau ds.
$$
If $\lambda=1$, then $c_0=0$ and if $|c_0|>N^*$, in view of
(\ref{e3.4}), one has
$$
\lambda c^{2}_{0}=(1-\lambda)c_0\sum_{i=1}^{m-2}\alpha_i
\int_0^{\xi_i}\frac{1}{q(s)}\int_s^{\infty}f(\tau,c_0,0)d\tau
ds<0,
$$
which contradicts $\lambda c^{2}_{0}\ge 0$. Thus, $\Omega_3$
is bounded.

If (\ref{e3.5}) holds, then let
$$
\Omega_3=\{x\in \ker L:\lambda x+(1-\lambda)JQNx=0,\; \lambda\in [0,1]\},
$$
similar to the above argument, we can show that $\Omega_3$ is
bounded.

In the following, we shall prove that all conditions of Lemma
\ref{lem2.4} are satisfied. Let $\Omega$ to be a bounded open subset
of $X$ such that $\cup_{i=1}^{3}\overline \Omega_{i}\subset \Omega$.
Then
by the above argument, we have\\
(1) $Lx\neq \lambda Nx$ for every $(x,\lambda)\in [(\mathop{\rm dom} L\setminus \ker L)\cap
\partial \Omega]\times (0,1)$;\\
(2) $Nx \not\in\mathop{\rm Im} L$ for every $x\in \ker
L\cap\partial\Omega$.

Lastly, we will prove that (3) of Lemma \ref{lem2.4} is
satisfied. Define
$$
H(x,\lambda)=\pm \lambda x+(1-\lambda)QNx.
$$
It is obvious that $H(x,\lambda)\neq0$ for every
$x\in \partial\Omega \cap \ker L$. Thus,
\begin{align*}
\deg (JQN|_{\ker L\cap \partial\Omega},\Omega\cap \ker L, 0)
&=\deg (H(\cdot,0), \Omega\cap \ker L,0)\\
&=\deg (H(\cdot,1), \Omega\cap \ker L,0)\\
&=\deg (\pm I, \Omega\cap \ker L,0)\neq 0.
\end{align*}
Then by Lemma \ref{lem2.4}, $Lx=Nx$ has at least one solution in
$\mathop{\rm dom}L\cap \overline \Omega$. In other words,
(\ref{e1.1})-(\ref{e1.2}) has at least one solution in
$C^1[0,\infty)$.
\end{proof}

\subsection*{Acknowledgements}
The author would like to express his sincere appreciation to the
anonymous referee  for his/her helpful comments in improving the
presentation and quality of this article.

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