% First Name: Nuket
% Family Name: Aykut Hamal
% First Name: Fulya
% Family Name: Yoruk Deren
\documentclass[reqno]{amsart}
\usepackage{hyperref}


\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 19, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/19\hfil Second-order boundary-value problems]
{Second-order boundary-value problems with
integral boundary conditions  on the real line}

\author[F. Yoruk Deren, N. Aykut Hamal\hfil EJDE-2014/19\hfilneg]
{Fulya Yoruk Deren, Nuket Aykut Hamal}  % in alphabetical order

\address{Fulya Yoruk Deren  \newline
Department of Mathematics, Ege University,
 35100 Bornova, Izmir, Turkey}
\email{fulya.yoruk@ege.edu.tr}

\address{Nuket Aykut Hamal \newline
Department of Mathematics, Ege University,
 35100 Bornova, Izmir, Turkey}
\email{nuket.aykut@ege.edu.tr}

\thanks{Submitted October 16, 2013. Published January 10, 2014.}
\subjclass[2000]{34B10, 39A10, 34B18, 45G10}
\keywords{Integral boundary conditions; Fixed-point theorem;
infinite interval}

\begin{abstract}
 This article shows the existence and multiplicity of nonnegative
 solutions for nonlinear boundary-value problems with integral boundary
 conditions on the whole line.
 The arguments are based upon the Krasnoselskii' s fixed point theorem
 of cone expansion-compression type. An example is  given to demonstrate
 our results.
\end{abstract}

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


\section{Introduction}

The theory of boundary-value problems on an infinite interval for
differential equations has become an important area of investigation in
recent years. There are many results about  the existence of positive
solutions on an infinite interval for boundary value problems.
We refer the reader to
\cite{Agarwal, Lian, Liu, Yuji, Yan, Yoruk, Yude, Yu, Zima}
 and the references therein.

 At the same time,  boundary value problems with integral boundary conditions
for ordinary differential equations represent a very interesting and important
class of problems. They constitute two, three, multi-point and nonlocal
boundary value problems as special cases. The existence results of
positive solutions for such problems have received a great deal of attention.
To identify a few, we refer the reader to \cite{Boucherif, Wang, Ge} and
the references therein.

To the author' s knowledge, there are relatively few papers available
for the boundary-value problems with integral boundary conditions
on the half line and the real line. (See \cite{Yuji, Yoruk,  Yu, Zhang b}).
 Yoruk and Hamal \cite{Yoruk} considered the following
 boundary-value problem with  integral boundary conditions on an infinite interval,
\begin{gather}
\frac{1}{p(t)}(p(t)x'(t))'+f(t,x(t),x'(t))=0, \quad t\in(0,\infty),
\label{e1.1}\\
\begin{gathered}
a_1 x(0)- b_1\lim_{t\to 0^{+}} p(t)x'(t)
=\int_0^{\infty}g_1(x(s))\psi(s)d s, \\
a_2 \lim_{t\to +\infty}x(t)+b_2\lim_{t\to +\infty}
p(t)x'(t)=\int_0^{\infty}g_2(x(s))\psi(s)ds.
\end{gathered} \label{e1.2}
\end{gather}
The authors showed the existence results of solutions by means of the
 Shauder fixed point theorem and the Leggett-Williams fixed point theorem.

In this article,  we  are interested in the existence and multiplicity
of nonnegative solutions for the following integral boundary-value problem
on the whole line
\begin{gather}
(p(t)x'(t))' +\lambda q(t) f(t,x(t),x'(t)) =0, \quad
 t\in \mathbb{R}, \label{e1.3}\\
\begin{gathered}
a_1 \lim_{t\to {-\infty}} { x(t)-b_1\lim_{t\to {-\infty}}
p(t)x'(t)}=\int_{-\infty}^{\infty}g_1(s,x(s),x'(s))\psi(s)d s,\\
a_2 \lim_{t\to +\infty}x(t)+b_2\lim_{t\to +\infty}
p(t)x'(t)=\int_{-\infty}^{\infty}g_2(s,x(s),x'(s))\psi(s)ds,
\end{gathered} \label{e1.4}
\end{gather}
 where $\lambda>0$ is a parameter,
$f,  g_1, g_2\in\mathcal{C}( \mathbb{R}\times [0,\infty)\times\mathbb{R},
[0,\infty))$,
$q, \psi\in\mathcal{C}(\mathbb{R},(0,\infty))$ and
 $p\in\mathcal{C}(\mathbb{R},(0,\infty))\cap\mathcal{C}^{1}(\mathbb{R})$.
Here, the values of $\int_{-\infty}^{+\infty}g_i(s,x(s),x'(s))ds$ $(i=1,2)$,
$\int_{-\infty}^{+\infty}\frac{d s}{p(s)}$ and $\sup_{s\in\mathbb{R}}\psi(s)$
are finite and  $a_1+ a_2> 0$, $b_i>0$ $(i=1,2)$ satisfying
$D =a_2b_1+a_1b_2+a_1a_2\int_{-\infty}^{+\infty}\frac{ds}{p(s)}>0$.


 The main features of our paper are as follows.
Firstly, compared with \cite{Yoruk}, we establish the existence results
 of solutions on $\mathbb{R}$ which expands the domain of definition of $t$
from a half line to the real line. Secondly, we investigate the existence
of solutions for the case $\lambda>0$, not $\lambda=1$ as in \cite{Yoruk}.

 The rest of this article is organized  as follows.
In Section 2, we represent some necessary lemmas that will be used to prove
our main results.
In Section 3, we apply the Krasnoselskii' s fixed point theorem to obtain
the existence and multiplicity of nonnegative solutions for
\eqref{e1.3}-\eqref{e1.4}. Finally, an example is given to illustrate
the main results.

 To the best of our knowledge, only a few papers deal with the existence
results of solutions for the boundary-value problem whose nonlinear
term $f$ involves $x$ and the first order derivative $x'$ explicitly,
especially  by means of the Krasnoselskii's fixed point theorem.
(See \cite{Zhang a}  and the references therein.)  So the main aim of this
work is to fill this gap.

\section{Preliminaries}

\label{sec2}  In this section, we present some preliminary results
and lemmas that will be used in the proof of our main results.
For convenience, we denote $\theta (t)$ and $\varphi (t)$ by
\begin{equation}
\theta(t)=b_1+a_1 \int_{-\infty}^{t}\frac{d\tau}{p(\tau)},\quad
\varphi(t)=b_2+a_2\int_{t}^{\infty}\frac{d\tau}{p(\tau)}\,.
\label{e2.1}
\end{equation}

\begin{lemma} \label{lemm2.1}
Under the conditions $D>0$ and $\int_{-\infty}^{\infty}\frac{ds}{p(s)}<+\infty$,
the  boundary-value problem
\begin{gather}
(p(t)x'(t))' + h(t) =0, \quad t\in \mathbb{R}, \label{e2.2}\\
\begin{gathered}
a_1 \lim_{t\to {-\infty}} { x(t)- b_1\lim_{t\to {-\infty}}
p(t)x'(t)}=\int_{-\infty}^{\infty}\sigma_1(s)ds, \\
a_2 \lim_{t\to +\infty}x(t)+b_2\lim_{t\to +\infty}
p(t)x'(t)=\int_{-\infty}^{\infty}\sigma_2(s)ds
\end{gathered}\label{e2.3}
\end{gather}
has a unique solution for any  $h,\sigma_1,\sigma_2\in L(\mathbb{R})$.
Furthermore, this unique solution can be expressed as
\begin{equation}
x(t)=\int_{-\infty}^{\infty}G(t,s)h(s)d s
+\frac{\varphi(t)}{D}\int_{-\infty}^{\infty}\sigma_1(s)ds
+\frac{\theta(t)}{D}\int_{-\infty}^{\infty}\sigma_2(s)ds. \label{e2.4}
\end{equation}
Here,
\begin{equation}
G(t,s)=\frac{1}{D}\begin{cases}
\theta(t)\varphi(s),& -\infty< t\leq s<+\infty; \\
\theta(s)\varphi(t),& -\infty<s\leq t<+\infty,
\end{cases} \label{e2.5}
\end{equation}
where $\theta(t)$ and $\varphi(t)$ are
given by \eqref{e2.1}.
\end{lemma}

\begin{remark}\label{rmk2.1} \rm
From \eqref{e2.5}, we can get the following properties of $G(t,s)$:
\begin{itemize}
\item[(1)] $G(t,s)$ is continuous on
$\mathbb{R}\times\mathbb{R}$.

\item[(2)] For any $s\in\mathbb{R}$, $G(t,s)$ is
continuous differentiable on $\mathbb{R}$, except $t=s$.

\item[(3)] $\frac{\partial G(t,s)}{\partial
t}\mid_{t=s^{+}} -\frac{\partial G(t,s)}{\partial
t}\mid_{t=s^{-}}=-\frac{1}{p(s)}$.

\item[(4)] For any $t,s\in\mathbb{R}$, $G(t,s)\leq G(s,s)$,
\begin{gather*}
\overline{G}(s):=\lim_{t\to +\infty} G(t,s)
=\frac{b_2}{D}\theta(s) \leq G(s,s)< +\infty,\\
\underline{G}(s):=\lim_{t \to
-\infty} G(t,s)=\frac{b_1}{D}\varphi(s) \leq G(s,s)< +\infty.
\end{gather*}
\item[(5)] For any $k>0$ real number, $t\in[-k,k]$
 and $s\in\mathbb{R}$, we have
\begin{equation}
G(t,s)\geq w G(s,s),\quad\text{where }
w=\frac{\min\{\varphi(k),\theta(-k)\}}
{\max\{\varphi(-\infty),\theta(\infty)\}}.\label{e2.6}
\end{equation}
It is obvious that $0<w<1$.
\end{itemize}
\end{remark}

We define the Banach space
\[
B =\{x\in\mathcal{C}'(\mathbb{R}): \lim_{t\to \mp\infty} x(t)<+\infty,
\lim_{t\to \mp\infty}x'(t)<+\infty \}
\]
 equipped with the norm
$\|x\| = \sup_{t\in\mathbb{R}}[|x(t)|+|x'(t)|]$
and the cone $P\subset B$ by
\[
P =\big\{x\in B : x(t)\geq 0\; \forall t\in\mathbb{R},\;
\min_{t\in[-k,k]} x(t)\geq w \sup_{t\in\mathbb{R}} |x(t)|,\;  k>0,\;
 [-k,k]\subset \mathbb{R} \big\}, %\label{e2.7}
\]
in which $w $ is given by  \eqref{e2.6}.
In this article, we need the following assumptions:
\begin{itemize}
\item[(H1)] $f, g_1, g_2 \in\mathcal{C}(\mathbb{R}\times
[0,\infty)\times\mathbb{ R},[0,\infty))$ and for any $t\in \mathbb{R}$ and
$i=1,2$, we have
\[
 u_2(t)h_3(x,y)\leq f(t,x,y)\leq u_1(t)h_3(x,y),\quad
 g_i(t,x,y)\leq v_i(t)h_i(x,y),
\]
 where $h_i\in\mathcal{C}([0,\infty)\times\mathbb{R},[0,\infty))$
$(i=1,2,3)$,
$u_i,v_i\in L(\mathbb{R},(0,\infty))$ $(i=1,2)$;
also there exists $0<\gamma_0<1$ such that $u_2(t)\geq\gamma_0 u_1(t)$.


\item[(H2)] $\int_{-\infty}^{\infty}G(s,s)q(s)u_i(s)ds<+\infty$, $(i=1,2)$.

\item[(H3)] $\psi:\mathbb{R}\to (0,\infty)$ is a continuous
function with $\sup_{s\in\mathbb{R}}\psi(s)<+\infty$.
\end{itemize}
Using the above assumptions, we  define the
operator $A$ on $P$ by
\begin{align*}
Ax(t)&=\lambda\int_{-\infty}^{\infty}G(t,s)q(s)f(s,x(s),x'(s))d
s+\frac{\varphi(t)}{D}\int_{-\infty}^{\infty}g_1(s,x(s),x'(s))\psi(s)ds\\
&\quad +\frac{\theta(t)}{D}\int_{-\infty}^{\infty}g_2(s,x(s),x'(s))\psi(s)ds,
\end{align*} %\label{e2.8}
where $G(t,s)$ is given by \eqref{e2.5}. Obviously, $x$ is a solution
of \eqref{e1.3}-\eqref{e1.4} if and only if $x$ is a fixed point of
the operator $A$.

\section{Main Results}

  In this section, we will apply the following
Krasnoselskii's fixed point theorem to establish the existence  and
multiplicity of nonnegative solutions for \eqref{e1.3}-\eqref{e1.4}.

\begin{lemma}[\cite{Guo}] \label{lem3.1} %{lem1.2}
Let $\mathcal{B}$ be a real Banach space and $P\subset B$ be a cone in $B$.
Assume that $\Omega_1, \Omega_2$ are open subsets of $B$ with
$0\in\Omega_1$ and $\overline{\Omega_1}\subset\Omega_2$ and let
$A: P\cap (\overline{\Omega_2}\setminus\Omega_1)\to P$ be a completely
continuous operator such that, either
\begin{itemize}
\item[(i)] $\|Ax\|\leq\|x\|$ for $x\in P\cap\partial\Omega_1$ and
$\|Ax\|\geq\|x\|$ for $x\in P\cap\partial\Omega_2$;

\item[(ii)] $\|Ax\|\geq\|x\|$ for $x\in P\cap\partial\Omega_1$ and
 $\|Ax\|\leq\|x\|$ for $x\in P\cap\partial\Omega_2$.
\end{itemize}
Then $A$ has at least one fixed point in
$P\cap (\overline{\Omega_2}\setminus\Omega_1)$.
\end{lemma}

\begin{lemma}\label{lem3.2} %{lem1.2}
Assume that {\rm (H1)--(H3)} are satisfied. Then the operator
$A:P\to P$ is completely continuous.
\end{lemma}

\begin{proof}  We assert that $A$ is a completely
continuous operator. To justify this, we first show that
$A: P\to B$ is well defined. Let $x\in P$,
then there exists $r>0$ such that $\|x\|\leq r$. From
condition (H1),  for any $t\in\mathbb{R}$, we have
\begin{gather*}
N_r:=\sup\{h_1(x,y):|x|+|y|\leq r\}<+\infty,\\
N'_r:=\sup\{h_2(x,y):|x|+|y|\leq r\}<+\infty,\\
M_r:=\sup\{h_3(x,y):|x|+|y|\leq r\}<+\infty.
\end{gather*}
Let $t_1,t_2\in\mathbb{R}$ with $t_1<t_2$, then it follows
from (H2) and (H3) that
\begin{align*}
&\lambda\int_{-\infty}^{\infty}|G(t_1,s)-G(t_2,s)|q(s)u_1(s)h_3(x(s),x'(s))ds\\
&\leq 2 \lambda M_r\int_{-\infty}^{\infty}G(s,s) q(s)u_1(s)d
s<+\infty
\end{align*}
 and
\begin{align*}
&\int_{-\infty}^{\infty}[v_1(s)h_1(x(s),x'(s))+v_2(s)h_2(x(s),x'(s))]\psi(s)ds\\
&\leq\int_{-\infty}^{\infty} [N_rv_1(s)+N'_rv_2(s)]\psi(s)d
s<+\infty.
\end{align*}
 Hence by the Lebesgue dominated convergence
theorem and the fact that $G(t,s)$ is continuous on $t$, we have
\begin{equation}
\begin{aligned}
&|(A x)(t_1)-(A x)(t_2)|\\
&\leq \lambda\int_{-\infty}^{\infty}|G(t_2,s)-G(t_1,s)|q(s)f(s,x(s),x'(s))ds\\
&\quad +\frac{|\varphi(t_2)-\varphi(t_1)|}{D}\int_{-\infty}^{\infty}
 g_1(s,x(s),x'(s))\psi(s)d s\\
&\quad +\frac{|\theta(t_2)-\theta(t_1)|}{D}\int_{-\infty}^{\infty}g_2(s,x(s),x'(s))
\psi(s)d s\\
&\leq 2 \lambda M_r\int_{-\infty}^{\infty}|G(t_2,s)-G(t_1,s)|q(s)u_1(s)ds\\
&\quad +\frac{1}{D}\int_{-\infty}^{\infty}[N_r |\varphi(t_2)-\varphi(t_1)|v_1(s)
+N'_r |\theta(t_2)-\theta(t_1)|v_2(s)]\psi(s)ds\\
&\quad \to 0,\quad\text{as }t_1\to t_2
\end{aligned} \label{e3.1}
\end{equation}
and
\begin{equation}
\begin{aligned}
&|(A x)'(t_1)-(A x)'(t_2)|\\
&\leq\frac{\lambda a_2}{D}|\frac{1}{p(t_1)}-\frac{1}{p(t_2)}|
\int_{-\infty}^{t_1} \theta(s)q(s)f(s, x(s),x'(s))d s\\
&\quad + \frac{\lambda a_1}{D p(t_1)}\int_{t_1}^{t_2}
\varphi(s)q(s)f(s, x(s),x'(s))d s
\\
&\quad +\frac{\lambda a_1}{D}|\frac{1}{p(t_1)}-\frac{1}{p(t_2)}|
\int_{t_2}^{\infty}\varphi(s)q(s)f(s, x(s),x'(s))d s
\\
&\quad +\frac{\lambda a_2}{Dp(t_2)}\int_{t_1}^{t_2}\theta(s)q(s)f(s, x(s),x'(s))
 d s
\\
&\quad +\frac{a_2}{D}|\frac{1}{p(t_1)}-\frac{1}{p(t_2)}|
\int_{-\infty}^{\infty}g_1(s,x(s),x'(s))\psi(s)ds
\\
&\quad +\frac{a_1}{D}|\frac{1}{p(t_1)}-\frac{1}{p(t_2)}|
\int_{-\infty}^{\infty}g_2(s,x(s),x'(s))\psi(s)ds
\\
&\leq \frac{\lambda a_2 M_{r}}{D}|\frac{1}{p(t_1)}
-\frac{1}{p(t_2)}| \int_{-\infty}^{t_1}\theta(s)q(s)u_1(s)d s
\\
&\quad +\frac{\lambda a_1 M_{r}}{Dp(t_1)} \int_{t_1}^{t_2}\varphi(s)q(s)u_1(s)d s
+\frac{\lambda a_1 M_{r}}{D}|\frac{ 1}{p(t_1)}-\frac{ 1}{p(t_2)}|
 \int_{t_2}^{\infty}\varphi(s)q(s)u_1(s)d s
\\
&\quad +\frac{\lambda a_2 M_{r} }{D p(t_2)}\int_{t_1}^{t_2}\theta(s)q(s)u_1(s)d s
+\frac{a_2 N_{r}}{D}|\frac{ 1}{p(t_1)}-\frac{ 1}{p(t_2)}|
\int_{-\infty}^{\infty}v_1(s)\psi(s)d s
\\
&\quad +\frac{a_1N'_{r}}{D}|\frac{ 1}{p(t_1)}-\frac{ 1}{p(t_2)}|
\int_{-\infty}^{\infty}v_2(s)\psi(s)d s
\to 0\quad\text{as }t_1\to t_2.
\end{aligned}     \label{e3.2}
\end{equation}
Thus, $Ax\in\mathcal{C}^{1}(\mathbb{R})$.

We can show that $Ax\in B$. Notice that
\begin{align*}
&\lim_{t\to +\infty}(Ax)(t)\\
&=\lambda\int_{-\infty}^{\infty} \overline{G}(s)q(s)f(s,x(s),x'(s))ds
+\frac{\varphi(+\infty)}{D} \int_{-\infty}^{\infty}g_1(s,x(s),x'(s))
\psi(s)d s\\
&\quad +\frac{\theta(+\infty)}{D}
\int_{-\infty}^{\infty}g_2(s,x(s),x'(s))\psi(s)d s<+\infty
\end{align*}
 and
\begin{align*}
&\lim_{t\to -\infty}(Ax)(t)\\
&=\lambda\int_{-\infty}^{\infty}\underline{G}(s)q(s)f(s,x(s),x'(s))d s
+\frac{\varphi(-\infty)}{D} \int_{-\infty}^{\infty} g_1(s,x(s),x'(s))\psi(s)ds
\\
&\quad +\frac{\theta(-\infty)}{D}
 \int_{-\infty}^{\infty} g_2(s,x(s),x'(s))\psi(s)d s<+\infty.
\end{align*}
In addition, we have
\begin{align*}
&|(Ax)'(t)|\\
&\leq\frac{1}{D}[\lambda\int_{-\infty}^{t}|\varphi'(t)|\theta(s)q(s)
f(s,x(s),x'(s))ds\\
&\quad +\lambda\int_{t}^{\infty}|\theta'(t)|\varphi(s)q(s)f(s,x(s),x'(s))ds
\\
&\quad +|\varphi'(t)|\int_{-\infty}^{\infty}g_1(s,x(s),x'(s))\psi(s)ds
+\theta'(t)\int_{-\infty}^{\infty}g_2(s,x(s),x'(s))\psi(s)ds]
\\
&\leq\frac{\max\{a_1,a_2\}}{\min\{b_1,b_2\}}
[\frac{\lambda}{p(t)}\int_{-\infty}^{\infty}G(s,s)q(s)f(s,x(s),x'(s))ds
\\
&\quad +\frac{\max\{\theta(+\infty),\varphi(-\infty)\}}{D p(t)}
\int_{-\infty}^{\infty}[g_1(s,x(s),x'(s))+g_2(s,x(s),x'(s))]\psi(s)ds]
\\
&\leq\frac{\max\{a_1,a_2\}}{\min\{b_1,b_2\}}
\sup_{t\in\mathbb{R}}\frac{1}{p(t)}
[\lambda M_r\int_{-\infty}^{\infty}G(s,s)q(s)u_1(s)ds
\\
&\quad +\frac{\max\{\theta(+\infty),\varphi(-\infty)\}}{D}
 \int_{-\infty}^{\infty}[N_r v_1(s)+N'_r v_2(s)]\psi(s)ds]
<+\infty,
\end{align*}
so, we have
$\lim_{t\to \mp\infty}(Ax)'(t)<+\infty$.
Hence, $A: P\to B$ is well defined.

Now, we prove that $A:P\to P$. It is obvious that $Ax(t)\geq 0$
for any $t\in\mathbb{R}$.
Let $x\in P$, then for all $t\in\mathbb{R}$, we have
\begin{align*}
&|Ax(t)|\\
&\leq\lambda\int_{-\infty}^{\infty}G(s,s)q(s)f(s,x(s),x'(s))ds\\
&\quad +\frac{\max\{\theta(+\infty),\varphi(-\infty)\}}{D}
\int_{-\infty}^{\infty}[g_1(s,x(s),x'(s))+g_2(s,x(s),x'(s))]\psi(s)ds.
\end{align*}
On the other hand, for any $k>0$, $t\in [-k,k]\subset\mathbb{R}$, we obtain
\begin{align*}
&|Ax(t)|\\
&\geq w \lambda\int_{-\infty}^{\infty}G(s,s)q(s)f(s,x(s),x'(s))ds\\
&\quad+\frac{\min\{\varphi(k),\theta(-k)\}}{D}
 \int_{-\infty}^{\infty}[g_1(s,x(s),x'(s))+g_2(s,x(s),x'(s))]\psi(s)ds\\
&= w\Big[ \lambda\int_{-\infty}^{\infty}G(s,s)q(s)f(s,x(s),x'(s))ds\\
&\quad +\frac{\max\{\theta(+\infty),\varphi(-\infty)\}}{D}
 \int_{-\infty}^{\infty}[g_1(s,x(s),x'(s))+g_2(s,x(s),x'(s))]\psi(s)ds\Big]\\
&\geq w \sup_{t\in\mathbb{R}}|Ax(t)|.
\end{align*}
Therefore, $A:P\to P$ is well defined.

Next we prove that $A:P\to P$ is  continuous.
Let $x_n,x\in P$ with $\|x_n-x\|\to 0$ as $n\to \infty$. We will show
that $\|Ax_n- A x\|\to 0$ as $n\to\infty$ in $P$.

From  (H1)--(H3) we obtain
\begin{align*}
&\lambda\int_{-\infty}^{\infty}G(s,s)q(s)| f(s,x_n(s),x'_n(s))
-f(s,x(s),x'(s))|d s\\
&\leq 2\lambda M_{r_0}\int_{-\infty}^{\infty}G(s,s)q(s)u_1(s)ds
<+\infty,
\\
&\int_{-\infty}^{\infty}|g_1(s,x_n(s), x'_n(s))-g_1(s,x(s),x'(s))|\psi(s)d s\\
&\leq  2N_{r_0}\int_{-\infty}^{\infty}v_1(s)\psi(s)d s
<+\infty,
\\
&\int_{-\infty}^{\infty}|g_2(s,x_n(s), x'_n(s))-g_2(s,x(s),x'(s))|\psi(s)ds\\
& \leq  2N'_{r_0}\int_{-\infty}^{\infty}v_2(s)\psi(s)ds
<+\infty,
\end{align*}
where $r_0>0$ is a real number such that
$r_0\geq\max_{n\in {\mathbb{N}-\{ 0\}}}\{\|x_n\|,\|x\|\}$.
Therefore,
\begin{align*}
&|(Ax_n)(t)-(A x)(t)|\\
&\leq \lambda\int_{-\infty}^{\infty}G(s,s)q(s)| f(s,x_n(s),x'_n(s))
-f(s,x(s),x'(s))|ds \\
&\quad +\frac{\varphi(-\infty)}{D}\int_{-\infty}^{\infty}|g_1(s,x_n(s),
 x'_n(s))-g_1(s,x(s),x'(s))|\psi(s)ds\\
&\quad +\frac{\theta(+\infty)}{D}\int_{-\infty}^{\infty}|g_2(s,x_n(s),
x'_n(s))-g_2(s,x(s),x'(s))|\psi(s)ds
\to 0 \quad\text{as } n\to\infty.
\end{align*}
Similarly, we can see that when $\|x_n-x\|\to 0$ as $n\to +\infty$,
\[
\lim_{n\to \infty}\sup_{t\in\mathbb{R}}|(Ax_n)'(t)-(Ax)'(t)| =0.
\]
This implies that  $A:P\to P$ is a continuous operator.

  Now, we show that $A$ maps bounded subsets into bounded subsets.
Let $D\subset P$ be bounded and $x\in D$, then there exists $R>0$
such that $\|x\|\leq R$, for any $x\in D$. Furthermore, for $t\in\mathbb{R}$,
we obtain
\begin{equation}
\begin{aligned}
|(A x)(t)|
&\leq \lambda\int_{-\infty}^{\infty}G(s,s)q(s)f(s,x(s),x'(s))ds \\
&\quad +\frac{\max\{\theta(+\infty),\varphi(-\infty)\}}{D}
\int_{-\infty}^{\infty}\big[g_1(s,x(s),x'(s))\\
&\quad +g_2(s,x(s),x'(s))\big]\psi(s)ds\\
&\leq \lambda  M_R \int_{-\infty}^{\infty}G(s,s)q(s)u_1(s)ds\\
&\quad +\frac{\max\{\theta(+\infty),\varphi(-\infty)\}}{D}
\int_{-\infty}^{\infty}[N_Rv_1(s)+N'_Rv_2(s)]\psi(s)ds
\end{aligned} \label{e3.3}
\end{equation}
and
\begin{equation}
\begin{aligned}
|(Ax)'(t)|
&\leq \frac{\max\{a_1,a_2\}}{\min\{b_1,b_2\}}
\sup_{t\in\mathbb{R}}\frac{1}{p(t)}
\Big[\lambda\int_{-\infty}^{\infty}G(s,s)q(s)f(s,x(s),x'(s))ds\\
&\quad +\frac{\max\{\theta(+\infty),\varphi(-\infty)\}}{D}
\int_{-\infty}^{\infty}\big[g_1(s,x(s),x'(s))\\
&\quad +g_2(s,x(s),x'(s))\big]\psi(s)ds\Big].
\end{aligned} \label{e3.4}
\end{equation}
Inequalities \eqref{e3.3} and \eqref{e3.4} imply
\begin{align*}
&|Ax(t)|+|(Ax)'(t)|\\
&\leq(1+\frac{\max\{a_1,a_2\}}{\min\{b_1,b_2\}}
\sup_{t\in\mathbb{R}}\frac{1}{p(t)})
\Big[\lambda\int_{-\infty}^{\infty}G(s,s)q(s)f(s,x(s),x'(s))ds\\
&\quad +\frac{\max\{\theta(+\infty),\varphi(-\infty)\}}{D}
\int_{-\infty}^{\infty}[g_1(s,x(s),x'(s))+g_2(s,x(s),x'(s))]\psi(s)ds\Big]
\\
&\leq (1+\frac{\max\{a_1,a_2\}}{\min\{b_1,b_2\}}
\sup_{t\in\mathbb{R}}\frac{1}{p(t)}
)\Big[ \lambda M_R\int_{-\infty}^{\infty}G(s,s)q(s)u_1(s)ds\\
&\quad + \frac{\max\{\theta(+\infty),\varphi(-\infty)\}}{D}
\int_{-\infty}^{\infty}[N_R v_1(s)+N'_R v_2(s)]\psi(s)d s
\Big].
\end{align*}
Hence, we obtain  $\sup_{t\in \mathbb{R}}[|Ax(t)|+|(Ax)'(t)|]<+\infty$; that is,
$A$ is uniformly bounded.

 Using the similar proof as the one for \eqref{e3.1} and \eqref{e3.2},
for any $N\in (0,\infty)$, $t,t_1\in[-N,N]$ and $x\in D$, we have
$\|Ax(t)-Ax(t_1)\|  \to 0$ as $t\to t_1$. Thus, $A D$ is equicontinuous
on any compact interval of $\mathbb{R}$.

By (H2), (H3) and the Lebesgue dominated convergence  theorem, we have
\begin{align*}
&|(A x)(t)-(A x)(+\infty)|\\
&\leq\lambda\int_{-\infty}^{\infty}|G(t,s)-\overline{G}(s)|q(s)f(s,x(s),x'(s))ds
\\
&\quad +\frac{|\varphi(t)-\varphi(+\infty)|}{D}
\int_{-\infty}^{\infty}g_1(s,x(s),x'(s))\psi(s)ds
\\
&\quad +\frac{|\theta(t)-\theta(+\infty)|}{D}
\int_{-\infty}^{\infty}g_2(s,x(s),x'(s))\psi(s)d s
\\
&\leq \lambda M_R\int_{-\infty}^{\infty}|G(t,s)-\overline{G}(s)|q(s)u_1(s)ds
\quad +\frac{1}{D}\int_{-\infty}^{\infty}
[N_{R} |\varphi(t)-\varphi(+\infty)|v_1(s)\\
&\quad +N'_{R}|\theta(t)-\theta(+\infty)|v_2(s)]\psi(s)ds
\to 0\quad\text{as }t\to \infty
\end{align*}
and
\begin{align*}
&|(Ax)'(t)-(Ax)'(\infty)|\\
&\leq\frac{1}{D}[|\frac{1}{p(t)}-\frac{1}{p(\infty)}|
\int_{-\infty}^{t}a_2 \theta(s)q(s)f(s,x(s),x'(s))ds
\\
&\quad +|\frac{1}{p(t)}-\frac{1}{p(\infty)}|\int_{t}^{\infty}
a_1 \varphi(s)q(s)f(s,x(s),x'(s))ds
\\
&\quad +\max\{a_1,a_2\}|\frac{1}{p(t)}-\frac{1}{p(\infty)}|
 \int_{-\infty}^{\infty}[g_1(s,x(s),x'(s))+g_2(s,x(s),x'(s))]\psi(s)ds]
\\
&\leq\frac{1}{D}[|\frac{1}{p(t)}-\frac{1}{p(\infty)}|
M_R \int_{-\infty}^{t}a_2 \theta(s)q(s)u_1(s)ds
\\
&\quad +|\frac{1}{p(t)}-\frac{1}{p(\infty)}|
M_R \int_{t}^{\infty}a_1 \varphi(s)q(s)u_1(s)ds
\\
&\quad +\max\{a_1,a_2\}|\frac{1}{p(t)}-\frac{1}{p(\infty)}|
\int_{-\infty}^{\infty}(N_R v_1(s)+N'_R v_2(s))\psi(s)ds]
\to 0\quad\text{as }t\to \infty.
\end{align*}
Therefore, $\{Ax: x\in D\}$ and $\{(Ax)': x\in D\}$
are equiconvergent at $+\infty$. Similarly, we can show that $A D$
is equiconvergent at $-\infty$.
Hence, we conclude that $A:P\to P$ is completely continuous.
Therefore, Lemma \ref{lem3.2} is proved.
\end{proof}

  For convenience, we denote
\begin{gather*}
A = w^2\gamma_0\int_{-k}^{k}G(s,s)q(s)u_1(s)ds, \quad
B = 2(1+\sup_{t\in\mathbb{R}}\frac{c}{p(t)})
\int_{-\infty}^{\infty}G(s,s)q(s)u_1(s)ds, \\
C = \frac{2(1+\sup_{t\in\mathbb{R}}\frac{c}{p(t)})
\max\{\theta(+\infty),\varphi(-\infty)\}}{D}
\int_{-\infty}^{\infty}(v_1(s)+v_2(s))\psi(s)ds,
\end{gather*}
where
\begin{equation}
c = \frac{\max\{a_1,a_2\}}{\min\{b_1,b_2\}}, \label{e3.5}
\end{equation}
$k>0$ is a real number and $w$ is defined by \eqref{e2.6}.

 In the next theorem, we also assume the following conditions
on $h_i(x,y)$ $(i=1,2,3)$.
\begin{itemize}
\item[(H4)] There exist numbers $0<r<R<+\infty$ such that for all
$t\in\mathbb{R}$,
\[
h_3(x,y)\geq  \frac{|x|+|y|}{\lambda A} \quad\text{for }
R\leq |x|+|y|<+\infty, \; 0\leq |x|+|y|\leq r.
\]

\item[(H5)] There exist numbers $0<r<p_1<R<+\infty$
$(r< \frac{Ap_1}{B})$   such that for all $t\in\mathbb{R}$,
\[
h_3(x,y)\leq \frac{p_1}{\lambda B}, \quad
h_i(x,y)\leq\frac{p_1}{ C}, \quad (i =1,2)\;    0\leq |x|+|y|\leq p_1.
\]
\end{itemize}

\begin{theorem}\label{thm3.1}
Assume that {\rm (H1)-(H5)} are satisfied. Then
\eqref{e1.3}-\eqref{e1.4} has at least two nonnegative solutions
$x_1$, $x_2$, $t\in\mathbb{R}$ such that
\[
0<\|x_1\|\leq p_1\leq\|x_2\|.
\]
\end{theorem}

\begin{proof}
Let $x\in P$ with $\|x\| =r$, then by (H4), we have
\begin{align*}
\|Ax\|&\geq |Ax(t)| \\
&\geq \lambda w \int_{-k}^{k} G(s,s)q(s)f(s,x(s),x'(s))ds\\
&\geq \lambda w \int_{-k}^{k} G(s,s)q(s)u_2(s)h_3(x(s),x'(s))ds\\
&\geq \frac{1}{\lambda A}\lambda w \int_{-k}^{k} G(s,s)q(s)u_2(s)[x(s)+|x'(s)|]ds\\
&\geq\frac{w^2 \gamma_0}{ A} \|x\| \int_{-k}^{k} G(s,s)q(s)u_1(s)ds
=  \|x\|.
\end{align*}
If we let $\Omega_1 =\{x\in B: \|x\|< r\}$, then
\begin{equation}
\|Ax\|\geq \|x\|\quad\text{for all  }x\in P\cap\partial\Omega_1.
\label{e3.6}
\end{equation}
 Further, let $x\in P$ with $\|x\| =p_1$. Then from  (H5), we obtain
\begin{align*}
\|Ax\|&\leq (1+\sup_{t\in\mathbb{R}}\frac{c}{p(t)})
\Big[\lambda\int_{-\infty}^{\infty}G(s,s)q(s)f(s,x(s),x'(s))ds\\
&\quad +\frac{\max\{\theta(+\infty),\varphi(-\infty)\}}{D}
\int_{-\infty}^{\infty}[g_1(s,x(s),x'(s))+g_2(s,x(s),x'(s))]\psi(s)ds\Big]\\
&\leq (1+\sup_{t\in\mathbb{R}}\frac{c}{p(t)})
\Big[\frac{p_1}{\lambda B} \lambda \int_{-\infty}^{\infty}G(s,s)q(s)u_1(s)ds\\
&\quad +\frac{p_1\max\{\theta(+\infty),\varphi(-\infty)\}}{C D}
\int_{-\infty}^{\infty}(v_1(s)+v_2(s))\psi(s)ds\Big]\\
&=p_1=\|x\|.
\end{align*}
Thus,
\begin{equation}
\|Ax\|\leq \|x\|\quad\text{for all  }x\in P\cap\partial\Omega_2, \label{e3.7}
\end{equation}
where $\Omega_2 =\{x\in B: \|x\|< p_1\}$.
Lemma \ref{lem3.1}, \eqref{e3.6} and \eqref{e3.7} imply that there exists a
fixed point $x_1$  in $P\cap(\overline{\Omega_2}\setminus\Omega_1)$
satisfying $r\leq\|x_1\|\leq p_1$.

  On the other hand, let $R_1 =R/w$ and $\Omega_3 =\{x\in B: \|x\|< R_1\}$.
Then $x\in P$ with $\|x\| =R_1$, $k\in(0,\infty)$, $t\in[-k,k]$ implies
\[
|x(t)|+|x'(t)|\geq w\|x\| =R\quad\text{for } t\in[-k,k].
\]
Therefore, from  (H4) again, we have
\begin{align*}
\|Ax\|&\geq \lambda w \int_{-k}^{k} G(s,s)q(s)f(s,x(s),x'(s))ds\\
&\geq \frac{1}{\lambda A}\lambda w \int_{-k}^{k} G(s,s)q(s)u_2(s)
[x(s)+|x'(s)|]ds \\
&\geq\frac{w^2\gamma_0}{ A} \|x\| \int_{-k}^{k} G(s,s)q(s)u_1(s)ds
 =  \|x\|.
\end{align*}
Therefore,
\begin{equation}
\|Ax\|\geq \|x\|\quad\text{for all }
x\in P\cap\partial\Omega_3.\label{e3.8}
\end{equation}
Lemma \ref{lem3.1}, \eqref{e3.7} and \eqref{e3.8} imply that there exists
a fixed point $x_2$  in $P\cap(\overline{\Omega_3}\setminus\Omega_2)$
satisfying $p_1\leq\|x_2\|\leq R_1$.

Both $x_1$ and $x_2$ are nonnegative solutions of  \eqref{e1.3}-\eqref{e1.4}
and $0<\|x_1\|\leq p_1\leq\|x_2\|$ holds.
\end{proof}

  In Theorem \ref{thm3.2}, we will assume the following conditions on
$h_i(x,y)$ $(i=1,2,3)$.
\begin{itemize}
\item[(H6)] There exist numbers $0<r<R<+\infty$ such that for all
$t\in\mathbb{R}$,
\[
h_3(x,y)\leq  \frac{|x|+|y|}{\lambda B}, \quad
h_i(x,y)\leq\frac{|x|+|y|}{ C}\quad  (i =1,2)
\]
for $0\leq |x|+|y|\leq r$ and $0\leq |x|+|y|\leq R$.

\item[(H7)] There exist numbers $0<r<p_2<R<+\infty$
$(r< \frac{A}{B}p_2)$ such that for all $t\in\mathbb{R}$,
\[
h_3(x,y)\geq \frac{|x|+|y|}{\lambda A}, \quad 0\leq |x|+|y|\leq p_2.
\]
\end{itemize}

\begin{theorem}\label{thm3.2}
Assume that {\rm (H1)-(H3), (H6), (H7)} are satisfied.
Then   \eqref{e1.3}-\eqref{e1.4} has at least two nonnegative solutions
$x_1$, $x_2$, $t\in\mathbb{R}$ such that
$0<\|x_1\|\leq p_2\leq\|x_2\|$.
\end{theorem}

\begin{proof}
For $x\in P$  with $\|x\| =r$, we have
\begin{align*}
\|Ax\|&\leq (1+\sup_{t\in\mathbb{R}}\frac{c}{p(t)})
\Big[\lambda\int_{-\infty}^{\infty}G(s,s)q(s)f(s,x(s),x'(s))ds\\
&\quad +\frac{\max\{\theta(+\infty),\varphi(-\infty)\}}{D}
\int_{-\infty}^{\infty}[g_1(s,x(s),x'(s))+g_2(s,x(s),x'(s))]\psi(s)ds\Big]\\
&\leq (1+\sup_{t\in\mathbb{R}}\frac{c}{p(t)})
\Big[\frac{r}{\lambda B} \lambda \int_{-\infty}^{\infty}G(s,s)q(s)u_1(s)ds\\
&\quad +\frac{\max\{\theta(+\infty),\varphi(-\infty)\}r}{CD}
\int_{-\infty}^{\infty}(v_1(s)+v_2(s))\psi(s)ds\Big]\\
&=r =\|x\|.
\end{align*}
  Hence,
\begin{equation}
\|Ax\|\leq \|x\|\quad \forall  x\in P\cap\partial\Omega_1,\label{e3.9}
\end{equation}
where  $\Omega_1 =\{x\in B: \|x\|< r\}$.

 On the other hand, let $x\in P$ with $\|x\| =p_2$, then for any
$t\in [-k,k]$, we have
\begin{align*}
\|Ax\|
&\geq \lambda w \int_{-k}^{k} G(s,s)q(s)f(s,x(s),x'(s))ds \\
&\geq \frac{1}{\lambda A}\lambda w \int_{-k}^{k}
  G(s,s)q(s)u_2(s)[x(s)+|x'(s)|]ds\\
&\geq\frac{w^2\gamma_0}{ A} \|x\| \int_{-k}^{k} G(s,s)q(s)u_1(s)ds
 =  \|x\|.
\end{align*}
Therefore, if we choose $\Omega_2 =\{x\in B: \|x\|< p_2\}$, then
\begin{equation}
\|Ax\|\geq \|x\|\quad\forall  x\in P\cap\partial\Omega_2.
\label{e3.10}
\end{equation}
Lemma \ref{lem3.1}, \eqref{e3.9} and \eqref{e3.10} imply that there exists
a fixed point $x_1$  in $P\cap(\overline{\Omega_2}\setminus\Omega_1)$
satisfying $r\leq\|x_1\|\leq p_2$.

Next set $\Omega_3 =\{x\in B: \|x\|< R\}$. Then $x\in P$ with $\|x\| =R$,
so by (H6), we have
\begin{align*}
\|Ax\|&\leq (1+\sup_{t\in\mathbb{R}}\frac{c}{p(t)})
\Big[\lambda\int_{-\infty}^{\infty}G(s,s)q(s)f(s,x(s),x'(s))ds\\
&\quad +\frac{\max\{\theta(+\infty),\varphi(-\infty)\}}{D}
\int_{-\infty}^{\infty}[g_1(s,x(s),x'(s))+g_2(s,x(s),x'(s))]\psi(s)ds\Big]\\
&\leq (1+\sup_{t\in\mathbb{R}}\frac{c}{p(t)})
\Big[\frac{R}{\lambda B} \lambda \int_{-\infty}^{\infty}G(s,s)q(s)u_1(s)ds\\
&\quad +\frac{R}{ C}\frac{\max\{\theta(+\infty),\varphi(-\infty)\}}{D}
\int_{-\infty}^{\infty}(v_1(s)+v_2(s))\psi(s)ds\Big]\\
&=R =\|x\|.
\end{align*}
Therefore,
\begin{equation}
\|Ax\|\leq \|x\|\quad\text{for all  }x\in P\cap\partial\Omega_3. \label{e3.11}
\end{equation}
Lemma \ref{lem3.1}, \eqref{e3.10} and \eqref{e3.11} imply that there exists
a fixed point $x_2$  in $P\cap(\overline{\Omega_3}\setminus\Omega_2)$
satisfying $p_2\leq\|x_2\|\leq R$.
Both $x_1$ and $x_2$ are nonnegative solutions of
\eqref{e1.3}-\eqref{e1.4}  and $0<\|x_1\|\leq p_2\leq\|x_2\|$ holds.
\end{proof}

  In the next theorem, we  assume the following condition on
$h_i(x,y)$ $(i=1,2,3)$.
\begin{itemize}
\item[(H8)] There exist numbers $0<r<R<+\infty$ 
$(r< AR/B)$ such that for all
$t\in\mathbb{R}$,
\begin{gather*}
h_3(x,y)\geq \frac{1}{\lambda A}[|x|+|y|], \quad 0\leq |x|+|y|\leq r,\\
h_3(x,y)\leq  \frac{R}{\lambda B}, \quad
h_i(x,y)\leq\frac{R}{ C}, \quad (i =1,2)\;  0\leq |x|+|y|\leq R.
\end{gather*}
\end{itemize}

\begin{theorem}\label{thm3.3}
Assume that {\rm (H1)-(H3), (H8)} are satisfied.
Then \eqref{e1.3}-\eqref{e1.4} has at least one nonnegative solution
$x(t)$, $t\in\mathbb{R}$ such that
\begin{gather*}
wr\leq x(t)\leq  R, \quad t\in[-k,k]; \\
0\leq x(t)\leq R, \quad t\in(-\infty,-k)\cup(k,\infty),\\
-R\leq x'(t)\leq R, \quad t\in\mathbb{R}.
\end{gather*}
\end{theorem}

  In the next theorem, we assume the following condition
on $h_i(x,y)$ $(i=1,2,3)$.
\begin{itemize}
\item[(H9)] There exist numbers $0<r<R<+\infty$ such that for all
$t\in\mathbb{R}$,
\begin{gather*}
h_3(x,y)\leq \frac{r}{\lambda B}, \quad h_i(x,y)\leq\frac{r}{ C}, \quad
(i =1,2)\quad   0\leq |x|+|y|\leq r,\\
h_3(x,y)\geq  \frac{R}{\lambda A}, \quad R\leq |x|+|y|<\infty.
\end{gather*}
\end{itemize}


\begin{theorem}\label{thm3.4}
Assume that {\rm (H1)-(H3), (H9)} are satisfied.
Then \eqref{e1.3}-\eqref{e1.4} has at least one nonnegative solution
$x(t)$, $t\in\mathbb{R}$ such that
\begin{gather*}
wr\leq x(t)\leq \frac{R}{w}, \quad t\in[-k,k]; \\
0\leq x(t)\leq\frac{ R}{w}, \quad t\in(-\infty,-k)\cup (k,\infty),\
- \frac{R}{w}\leq x'(t)\leq  \frac{R}{w}, \quad t\in\mathbb{R}.
\end{gather*}
\end{theorem}

The proofs of Theorem \ref{thm3.3} and \ref{thm3.4} are similar to those of
Theorem \ref{thm3.1} and \ref{thm3.2}. So, they are omitted.

\begin{remark} \label{rmk3.1} \rm
If
\[
\lim_{|x|+|y|\to 0^{+}}\frac{h_i(x,y)}{|x|+|y|}=0\quad (i =1,2,3)
\]
and
\[
\lim_{|x|+|y|\to \infty} \frac{h_3(x,y)}{|x|+|y|}  =\infty
\]
for all $t\in\mathbb{R}$, then (H9) will be satisfied for $r>0$ sufficiently
small and $R>0$ sufficiently large.
\end{remark}

\subsection*{Example}
Consider the second-order integral boundary-value problem
\begin{gather}
((1+t^{2})x'(t))'+\lambda\frac{[x(t)+|x'(t)|]^{2}}{t^2+1} =0,
\quad t\in\mathbb{R}, \label{e3.12}\\
\begin{gathered}
\lim_{t\to -\infty}x(t)- \lim_{t\to -\infty}(1+t^2) x'(t)
=\int_{-\infty}^{\infty}\frac{[x(t)+|x'(t)|]^{2}}{(t^2+1)^2}  dt,\\
\lim_{t\to +\infty}(1+t^2) x'(t) =0.
\end{gathered} \label{e3.13}
\end{gather}
Here, $p(t) =1+t^2$, $q(t) =1$, $a_1 =1$, $a_2 =0$, $b_1 =b_2 =1$,
$\psi(t) =\frac{1}{1+t^2}$,
\[
f(t,x(t),y(t)) =g_1(t,x(t),y(t))=\frac{[x(t)+|y(t)|]^{2}}{t^2+1}, \quad
g_2(t,x,y) =0.
\]
 It is obvious that
$f, g_1, g_2 \in\mathcal{C}(\mathbb{R}\times [0,\infty)\times
\mathbb{ R},[0,\infty))$. Set $h_1(x,y)=h_3(x,y)=(x+|y|)^2$,
$h_2(x,y)=0$, $u_1(t)=v_1(t)=\frac{1}{t^2+1}$, and $v_2(t)=0$.
It is clear that $h_i\in\mathcal{C}([0,\infty)\times\mathbb{R},[0,\infty))$
$(i=1,2,3)$, $u_1,v_1\in L(\mathbb{R},(0,\infty))$ and
\[
\int_{-\infty}^{\infty}G(s,s)q(s)u_1(s)ds
=\frac{\pi(\pi+2)}{2} <+\infty\quad  (i=1,2).
\]
 In addition to this, it is easy to see that
\begin{gather*}
\lim_{|x|+|y|\to 0^{+}}\frac{h_i(x,y)}{|x|+|y|}  =0, \quad (i =1,2,3),\\
\lim_{|x|+|y|\to \infty} \frac{h_3(x,y)}{|x|+|y|}  =\infty
\end{gather*}
for all $t\in\mathbb{R}$; that is, condition (H9) is satisfied
for $r>0$ sufficiently small and $R>0$ sufficiently large.
Hence, by Remark \ref{rmk3.1} and Theorem \ref{thm3.4}, the boundary-value problem
\eqref{e3.12}-\eqref{e3.13} has at least one nonnegative solution.

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