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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 135, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/135\hfil Positive solutions for a system]
{Positive solutions for a system of second-order boundary-value problems
involving first-order derivatives}

\author[K. Wang, Z. Yang \hfil EJDE-2012/135\hfilneg]
{Kun Wang, Zhilin Yang}  % in alphabetical order

\address{Kun Wang \newline
Department of Mathematics \\
Qingdao Technological University\\
Qingdao, 266033, China}
\email{wangkun880304@163.com}

\address{Zhilin Yang \newline
Department of Mathematics \\
Qingdao Technological University \\
Qingdao, 266033,  China}
\email{zhilinyang@sina.com}

\thanks{Submitted May 31, 2012. Published August 17, 2012.}
\subjclass[2000]{34B18, 45G15, 45M20, 47H07, 47H11}
\keywords{System of second-order boundary-value problems;
positive solution; \hfill\break\indent  first-order derivative;
fixed point index; $\mathbb{R}_+^2$-monotone matrix; concave function}

\begin{abstract}
 In this article we study  the existence and multiplicity of positive
 solutions for the system of second-order boundary value problems
 involving first order derivatives
 \begin{gather*}
 -u''=f(t, u, u', v, v'),\\
 -v''=g(t, u, u', v, v'),\\
 u(0)=u'(1)=0,\quad v(0)=v'(1)=0.
 \end{gather*}
 Here $f,g\in C([0,1]\times \mathbb{R}_+^{4},\mathbb{R}_+)
 (\mathbb{R}_+:=[0,\infty))$.
 We use fixed point index theory to establish our main results based on a priori
 estimates achieved by utilizing Jensen's integral inequality for concave functions
 and $\mathbb{R}_+^2$-monotone matrices.
\end{abstract}

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


\section{Introduction}
In this article we study  the existence and multiplicity of positive
solutions for the system of second-order boundary value problems
involving first order derivatives
\begin{equation}\label{second-order}
\begin{gathered}
-u''=f(t, u, u', v, v'),\\
-v''=g(t, u, u', v, v'),\\
u(0)=u'(1)=0,\quad v(0)=v'(1)=0,
\end{gathered}
\end{equation}
where $f\in C([0,1]\times \mathbb{R}_+^{4},\mathbb{R}_+)$ and
$g\in C([0,1]\times \mathbb{R}_+^{4},\mathbb{R}_+)$. By a positive solution
of \eqref{second-order}, we mean a pair of functions
$(u,v)\in C^{2}[0,1]\times C^{2}[0,1]$ that solve \eqref{second-order} and
satisfy $u(t)\geq 0$, $v(t)\geq 0$ for all $t\in [0,1]$,
with at least one of them positive on $(0,1]$.

Boundary-value problems for systems of nonlinear second-order
ordinary differential equations arise from physics, biology,
chemistry, and other applied sciences,  and, as a result,     play
an important role in both theory and application. Recently, there
are many articles in this direction. We refer the reader to
\cite{Caglar,Hu,Infante,Kong,PingKang,Lishan,Liuw,Lu,thompson,Zhongli,Xi,%
Jinbao,Yang,Youming} and the references cited therein. It should
remarked that in the works cited above, only a few of them involve
first-order derivatives in their nonlinearities.

 In \cite{Wang}, the authors study the existence and multiplicity  of positive
 solutions for the system
 \begin{gather*}
 \begin{aligned}
 &(-1)^mw^{(2m)}\\
 &=f(t, w, w',-w''',\dots, (-1)^{m-1}w^{(2m-1)}, z, z',-z''',\dots,
 (-1)^{n-1}z^{(2n-1)}), \end{aligned}\\
 \begin{aligned}
 &(-1)^nz^{(2n)}\\
 &=g(t, w, w',-w''',\dots, (-1)^{m-1}w^{(2m-1)}, z, z',-z''',\dots,
 (-1)^{n-1}z^{(2n-1)}), \end{aligned}\\
 w^{(2i)}(0)=w^{(2i+1)}(1)=0\quad (i=0,1,\dots, m-1),\\
 z^{(2j)}(0)=z^{(2j+1)}(1)=0\quad (j=0,1,\dots, n-1).
 \end{gather*}
The hypotheses imposed on the nonlinearities $f$ and $g$ are
formulated in terms of two linear functions $h_1(x)$ and $h_2(y)$.
The main results in \cite{Wang} are established by using fixed point
index theory  based on a priori estimates of positive solutions
achieved by utilizing new integral inequalities and  nonnegative
matrices.

In \cite{Yang6}, motivated by \cite{Yang3}, Yang and Kong studied
the system of second-order boundary value problems involving
first-order derivatives
\begin{equation}\label{Yang}
 -u''_{i}=f_i(t,u_1,u_1',\dots,u_n,u_n'), u_i(0)=u'_i(1)=0,i=1,\dots,n.
\end{equation}
To obtain the a priori estimates of positive solutions, the authors
develop  some integral identities and inequalities so  that the main
conditions imposed on $f_i'$s in \cite{Yang6}  can be formulated in
terms of   simple linear functions of the form
$g_i(x_1,\dots,x_{2n}):=\sum^{n}_{i=1}a_i(x_{2i-1}+2x_{2i})$. More
precisely, for example, (H2) in  \cite{Yang6} states that there
exist a nonnegative matrix $A = (a_{ij})_{n\times n}$ and a constant
$c> 0$ such that the matrix $A- I$ is an $\mathbb{R}^n_ +$-monotone
matrix and
\[
f_i(t, x)\geq \sum_{j=1}^n a_{ij}(x_{2j-1} + 2x_{2j}) -c
\]
for all $(t, x) \in  [0, 1] \times  \mathbb{R}^{2n}_+$, $i = 1,\dots, n$.


Motivated by \cite{Wang,Yang6,Yang3}, in this paper, we study the
existence and multiplicity of positive solutions for
\eqref{second-order}. We use fixed point index theory to establish
our main results based on a priori estimates of positive solutions
for some associated  problems, generalizing the corresponding ones
for the  single boundary value problem
\[
 -u''=f(t,u,u'),\quad u(0)=u'(1)=0
\]
in \cite{Yang3}. Our  generalizations are not routine, as our
conditions imposed on the nonlinearities $f$ and $g$, unlike these
in \cite{Yang3}, involve both linear functions on $\mathbb{R}_+^4$
and  concave functions on $\mathbb{R}_+ $; these functions  describe
how the nonlinearities $f,g$ grow and enable us  to treat the three
cases of them: one with both superlinear, one with both sublinear
and the last with one superlinear and the other sublinear. Also, it
is of interest to note that, for nonnegative constants
$p,q,\xi,\eta$ and nonnegative concave functions $\varphi$, $\psi$,
we  have to prove  the ratio
\[
\frac{\int_0^1(pu(t)+2qu'(t))\varphi(t)dt}
{\int_0^1(\xi u(t)+2\eta u'(t))\psi(t)dt}
\]
is bounded away from  both  0 and $\infty$
(see Lemma \ref{lem2.2} below for more details).
This is a great difference between this article and
\cite{Wang,Yang6}.

We use fixed point index theory to establish our main results based
on a priori estimates of positive solutions achieved by utilizing
Jensen's integral equality for concave functions and  $\mathbb{R}_+^2$-monotone
matrices. More precisely, Jensen's inequality is mainly  applied to
 derive the boundedness of weighted integrals of positive solutions for some
problems associated  to \eqref{second-order}, whereas $\mathbb{R}_+^2$-monotone
matrices  are employed  to solve  systems of
inequalities resulting from some weighted integrals and thereby
achieve the boundedness of associated weighted integrals.

This article is organized as follows. Section 2 contains  some
preliminary results, including two new integral inequalities
and a new integral identity.
 Our main  results, namely Theorems \ref{thm3.1}--\ref{thm3.3}, are stated
and proved in Section 3. Finally, in Section 4, we presented four
examples of nonlinearities to illustrate our main results.

\section{Preliminaries}
 Let $E:=C^1([0,1],\mathbb{R})$ and
\[
P:=\{u\in E: u(t)\geq 0,u'(t)\geq 0, \forall t\in
[0,1]\}, \|u\|:=\max\{\|u\|_{0},\|u'\|_{0}\},
\]
where
$\|u\|_{0}:=\max\{|u(t)|:t\in[0,1]\}$. Clearly, $(E, \|\cdot\|)$ is
a real Banach space and  $P$ is a cone in $E$. For $(u,v)\in E^2$,
let
\[
\|(u,v)\|:=\max\{\|u \|, \|v\|\}.
\]
Then $E^2$ is also a real Banach space under the above norm and
$P^2$ is a cone in $E^2$.

Let $k(t,s):=\min\{t,s\}$ and
\[
(Tu)(t):=\int_0^1k(t,s)u(s)ds.
\]
Then $T: E\to E$ is a completely continuous, positive, linear
operator, with  the spectral radius $r(T)=\frac{4}{\pi^2}$ and
\begin{equation}\label{varphi}
(T\varphi)(s)=\int_0^1k(t,s)\varphi(t)dt=r(T)\varphi(s)
\end{equation}
where $\varphi(t):= \sin\frac{\pi}{2}t$.

In our setting,  problem \eqref{second-order} is equivalent to the
system of nonlinear integral equations
\begin{equation}\label{integro-ordinary(2)}
\begin{gathered}
u(t)=\int_0^1k(t,s)f(s, u(s),u'(s),v(s),v'(s))ds,\\
v(t)=\int_0^1k(t,s)g(s, u(s),u'(s),v(s),v'(s))ds.\\
\end{gathered}
\end{equation}
Define the operators $A_{i}(i=1,2):P^2\to P$ and
$A: P^2 \to P^2$ by
\begin{gather*}
 A_1(u,v)(t):=\int_0^1k(t,s)f(s, u(s),u'(s),v(s),v'(s))ds,\\
 A_2(u,v)(t):=\int_0^1k(t,s)g(s, u(s),u'(s),v(s),v'(s))ds,\\
 A(u,v)(t):=(A_{1}(u,v),A_{2}(u,v)).
\end{gather*}
Now $f\in C([0,1]\times \mathbb{R}_+^{4},\mathbb{R}_+)$ and
$g\in C([0,1]\times \mathbb{R}_+^{4},\mathbb{R}_+)$ imply that $A_{i}$ and
$A$ are completely continuous operators. Clearly,  the existence of
positive solutions for \eqref{second-order} is equivalent to that of
positive fixed points of $A: P^2\to P^2$.

To establish the priori estimates of positive solutions for some
problems associated with \eqref{second-order}, we need  two
transcendental equations(see \cite{Yang3}).

For any $\xi>\eta>0$, let $\mu(\xi,\eta)\in(1/\xi,1/\eta)$ denote
the minimal positive solution of the transcendental equation
\begin{equation}\label{transcendental equation1}
\eta\mu\sin\sqrt{\xi\mu-\eta^2\mu^2}-\sqrt{\xi\mu
-\eta^2\mu^2}\cos\sqrt{\xi\mu-\eta^2\mu^2}=0.
\end{equation}
Also, for any $\eta>\xi>0$, let $\nu(\xi,\eta)\in(1/\eta,1/\xi)$
denote the unique solution of the transcendental equation
\begin{equation}\label{transcendental equation2}
\eta\nu\sinh\sqrt{\eta^2\nu^2-\xi\nu}-\sqrt{\eta^2\nu^2
-\xi\nu}\cosh\sqrt{\eta^2\nu^2-\xi\nu}=0.
\end{equation}
in $(1/\eta,\infty)$. Let
\begin{equation}\label{e-function}
\varphi_{\xi,\eta}(t):=\begin{cases}
\frac{\pi}{2}\sin\frac{\pi t}{2},&\xi>0,\eta=0,\\
te^t ,&\xi=\eta>0,\\
\frac{\xi\mu(\xi,\eta)}{\sqrt{\xi\mu(\xi,\eta)-\eta^2\mu^2(\xi,\eta)}}
\exp(\eta\mu(\xi,\eta)t)\\
\times \sin(\sqrt{\xi\mu(\xi,\eta)-
\eta^2\mu^2(\xi,\eta)}t),&\xi>\eta>0,\\
\frac{\xi\nu(\xi,\eta)}{\sqrt{\eta^2\nu^2(\xi,\eta)-\xi\nu(\xi,\eta)}}
\exp(\eta\nu(\xi,\eta)t)\\
\times \sinh(\sqrt{\eta^2\nu^2(\xi,\eta)
-\xi\nu(\xi,\eta)}t), &\eta>\xi>0,\\
\end{cases}
\end{equation}
\begin{equation}\label{lambda}
\lambda(\xi,\eta):=\begin{cases}
\frac{\pi^2}{4\xi},& \xi>0,\eta=0\\
\frac{1}{\xi},& \xi=\eta>0,\\
\mu(\xi,\eta),& \xi>\eta>0,\\
\nu(\xi,\eta),& \eta>\xi>0.\\
\end{cases}
\end{equation}
for all $\xi>0,\eta\geq0$. Direct calculation shows
\begin{equation}\label{transcendental equation3}
\int_0^1\varphi_{\xi,\eta}(t)dt=1.
\end{equation}

\begin{lemma} \label{lem2.1}
Suppose $\psi\in C([0,1],\mathbb{R}_+)$ is
not identically vanishing on $[0,1]$,  and $v\in C([0,1],\mathbb{R}_+)$ is a
concave function. Let
$\varrho(\psi):=\int_0^1t\psi(t)dt>0$. Then we have
\begin{equation}\label{estimate1}
\int_0^1\psi(t)v(t)dt\geq v(1)\varrho(\psi).
\end{equation}
\end{lemma}
\begin{proof}
 By the concavity of $v$ and the nonnegativity of
$\psi$, we have
\[
  \int_0^1\psi(t)v(t)dt=\int_0^1\psi(t)v(t\cdot1+(1-t)\cdot0)dt
\geq v(1)\int_0^1t\psi(t)dt
  =v(1)\varrho(\psi).
\]
This completes the proof.
\end{proof}

Denote
\[
P_0:=\{u\in P: u\text{ is concave on } [0,1],u(0)=u'(1)=0\}.
 \]

\begin{lemma} \label{lem2.2}
 Let $\xi_i>0, \eta_i\geq0, \varphi_{(\xi_i,\eta_i)}(i=1,2,3)$ be defined by
 \eqref{e-function}.
Define
\[
\beta(\xi_1,\eta_1,\xi_2,\eta_2,\xi_3,\eta_3):=\sup_{u\in
P_0\setminus\{0\}}\frac{\int_0^1(\xi_1u(t)+2\eta_1u'(t))\varphi_{\xi_2,\eta_2}(t)dt}
{\int_0^1(\xi_3u(t)+2\eta_3u'(t))\varphi_{\xi_3,\eta_3}(t)dt}.
\]
 Then
$0<\beta(\xi_1,\eta_1,\xi_2,\eta_2,\xi_3,\eta_3)<\infty$.
\end{lemma}

\begin{proof}  If  $u\in P_0$, then by  Lemma \ref{lem2.1}, we have
\begin{equation}\label{integro-eqution2}
\begin{aligned}
 &\int_0^1(\xi_3u(t)+2\eta_3u'(t))\varphi_{\xi_3,\eta_3}(t)dt\\
  &\geq\int_0^1\xi_3u(t)\varphi_{\xi_3,\eta_3}(t)dt
 =\xi_3\int_0^1u(t)\varphi_{\xi_3,\eta_3}(t)dt\\
  &\geq u(1)\xi_3\varrho(\varphi_{\xi_3,\eta_3})
 =\xi_3\|u\|_{0}\varrho(\varphi_{\xi_3,\eta_3})\\
\end{aligned}
\end{equation}
and \begin{equation}\label{second}
\begin{aligned}
&\int_0^1(\xi_1u(t)+2\eta_1u'(t))\varphi_{\xi_2,\eta_2}(t)dt\\
&=\xi_1\int_0^1u(t)\varphi_{\xi_2,\eta_2}(t)dt+2\eta_1\int_0^1u'(t)\varphi_{\xi_2,\eta_2}(t)dt\\
&\leq \xi_1\|\varphi_{\xi_2,\eta_2}\|_{0}\int_0^1u(t)dt+2\eta_1\|\varphi_{\xi_2,\eta_2}\|_0\int_0^1u'(t)dt\\
&\leq \xi_1\|\varphi_{\xi_2,\eta_2}\|_{0}\|u\|_{0}+2\eta_1\|\varphi_{\xi_2,\eta_2}\|_{0}u(1)\\
&=(\xi_1+2\eta_1)\|\varphi_{\xi_2,\eta_2}\|_{0}\|u\|_{0}.
\end{aligned}
\end{equation}
Combining  \eqref{integro-eqution2} and \eqref{second}, we obtain
\[
\beta(\xi_1,\eta_1,\xi_2,\eta_2,\xi_3,\eta_3)
\leq\frac{(\xi_1+2\eta_1)\|\varphi_{\xi_2,\eta_2}\|_{0}}
{\xi_3\varrho(\varphi_{\xi_3,\eta_3})}<\infty.
\]
This completes the proof.
\end{proof}

\begin{lemma} \label{lem2.3}
 If $ u\in C^{2}[0,1]$, $u(0)=u'(1)=0$, $\xi>0$, $\eta\geq 0$, then
\begin{equation}\label{integro-eqution1}
\int_0^1-u''(t)\varphi_{\xi,\eta}(t)dt=\lambda(\xi,\eta)\int_0^1(\xi u(t)+2\eta
u'(t))\varphi_{\xi,\eta}(t)dt,
\end{equation}
where $\varphi_{\xi,\eta}$ and $\lambda(\xi,\eta)$ are defined by
\eqref{e-function} and \eqref{lambda} respectively.
\end{lemma}

 \begin{proof}
We just prove \eqref{integro-eqution1} in the case
$\xi>\eta>0$; the remaining  cases can be proved in the same way.
Let
\[
a:=\eta\mu(\xi,\eta),\quad
b:=\sqrt{\xi\mu(\xi,\eta)-\eta^2\mu^2(\xi,\eta)}.
\]
Then $\varphi_{\xi,\eta}(t)=\frac{\xi a}{\eta b}e^{at}\sin bt$, and
\begin{equation}\label{lemma2.2.1}
a^2+b^2=\xi\mu(\xi,\eta),a\sin b-b\cos b=0.
\end{equation}
Integrate by parts over $[0,1]$  and use \eqref{lemma2.2.1} to
obtain
\begin{equation}\label{lemma2.2.2}
\begin{aligned}
&\int_0^1-u''(t)\varphi_{\xi,\eta}(t)dt \\
&=\frac{\xi a}{\eta b}\int_0^1-u''(t)e^{at}\sin btdt\\
&=\frac{\xi a}{\eta b}\int_0^1u'(t)e^{at}(a\sin bt+b\cos bt)dt\\
&=\mu(\xi,\eta)\int_0^12\eta u'(t)\varphi_{\xi,\eta}(t)dt
+\frac{\xi a}{\eta b}\int_0^1u'(t)e^{at}(b\cos bt-a\sin bt)dt\\
&=\mu(\xi,\eta)\int_0^1(\xi u(t)+2\eta
u'(t))\varphi_{\xi,\eta}(t)dt.
\end{aligned}
\end{equation}
 This completes the proof.
\end{proof}

\begin{lemma}[\cite{Guo}] \label{lem2.4}
 Let $E$ be a real Banach space and $P$ a cone on $E$. Suppose that
$\Omega\subset E$ is a bounded open set and that
 $T:\overline{\Omega}\bigcap P\to
P$ is a completely continuous operator. If there exists
$w_0\in P\backslash \{0\}$ such that
\[
w-Tw\neq \lambda w_0, \forall \lambda\geq 0,
\omega\in\partial\Omega\cap P,
\]
then $i(T,\Omega\bigcap P,P)=0$,
where $i$ indicates the fixed point index on $P$.
\end{lemma}

\begin{lemma}[\cite{Guo}] \label{lem2.5}
 Let $E$ be a real Banach space and $P$ a cone on $E$. Suppose that
$\Omega\subset E$ is a bounded open set with $0\in\Omega$ and that
 $T:\overline{\Omega}\cap P\to P$ is a completely continuous operator. If
\[
  w-\lambda Tw\neq0, \forall \lambda\in[0,1],  \quad w\in\partial\Omega\cap P,
\]
then $i(T,\Omega\cap P,P)=1$.
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.4]{Yang1}}] \label{lem2.6}
If  $p$ is  concave  on $[d,\infty)$, with
$\lim_{y\to\infty}p(y)/y\geq 0$, then $p$ is increasing on $[d,\infty)$ and
\begin{equation}\label{concave}
p(y+z-d)\leq p(y)+p(z)-p(d)
\end{equation}
for all $y,z\in [d,\infty)$.
\end{lemma}

\section{Existence of positive solutions for \eqref{second-order}}

\noindent\textbf{Definition} A real matrix $B$ is said to be nonnegative
if all elements of $B$ are nonnegative.

\noindent\textbf{Definition} (see \cite[p.112]{Berman})
 A real square matrix $M=(m_{ij})_{2\times 2}$ is called $\mathbb{R}_+^2$-monotone,
if for any column vector $x\in\mathbb{R}^2$,
$Mx\in\mathbb{R}_+^2\Rightarrow x\in\mathbb{R}_+^2$.

For simplicity, we denote by $x:=(x_1,x_2,x_3,x_4)\in\mathbb{R}_+^4$
and $I_\rho:=[0,\rho]$ for $\rho>0$. Now we list our hypotheses on
$f$ and $g$.
\begin{itemize}
\item[(H1)] $f, g\in C([0,1]\times\mathbb{R}_+^{4},\mathbb{R}_+)$.

\item[(H2)] There exist $p\in C(\mathbb{R}_+,\mathbb{R}_+)$ and $q\in
C(\mathbb{R}_+,\mathbb{R}_+)$ such that:
(1) $p$ is concave;
(2) There are two constants $c>0$ and $\mu_1>1$ such that
 \[
   f(t,x)\geq p(x_3)-c, g(t,x)\geq q(x_1)-c,\quad
 \forall(t,x)\in[0,1]\times\mathbb{R}_+^4,
\]
and
   \[
p(q(t))\geq \frac{\pi^4\mu_1}{16}t-c, \quad \forall t\in\mathbb{R}_+.
\]

\item[(H3)] For every $N>0$, there exist two functions
$\Phi_N, \Psi_N\in C(\mathbb{R}_+,\mathbb{R}_+)$ such that
\[
  f(t,x)\leq\Phi_N(x_{2}+x_{4}),\quad
  g(t,x)\leq\Psi_N(x_{2}+x_{4})
\]
for all $x\in I_N\times\mathbb{R}_+\times I_N\times\mathbb{R}_+$,
$t\in [0,1]$, and
\[
   \int_0^{\infty}\frac{\tau
   d\tau}{\Phi_N(\tau)+\Psi_N(\tau)+\delta}=\infty
\]
for all $\delta>0$.

\item[(H4)] There are  constants $a_{i}>0$, $b_{i}\geq0$, $c_{i}>0$,
$d_{i}\geq 0$  $(i=1,2)$ and $r>0$ such that
\[
\begin{pmatrix}
      f(t,x) \\
      g(t,x)
\end{pmatrix}
\leq
\begin{pmatrix}
      a_1 & 2b_1 & c_1 &2d_1 \\
      a_2 & 2b_2 & c_2 & 2d_2
\end{pmatrix}
\begin{pmatrix}
         x_1\\
         x_2 \\
         x_3 \\
         x_4
\end{pmatrix}
\]
for all $(t,x)\in[0,1]\times I_r^4$ and the matrix
\[
B_1:=\begin{pmatrix}\lambda(a_1,b_1)-1,&-\beta(c_1,d_1,a_1,b_1,c_2,d_2)
\\ -\beta(a_2,b_2,c_2,d_2,a_1,b_1),&
\lambda(c_2,d_2)-1\end{pmatrix}
\]
is an $\mathbb{R}_+^2$-monotone matrix, where the entries
 $\beta(c_1,d_1,a_1,b_1,c_2,d_2)$ and
$\beta(a_2,b_2,c_2,d_2,a_1,b_1)$ are defined as in Lemma \ref{lem2.2}.

\item[(H5)] There exist $\widetilde{p}\in C(\mathbb{R}_+,\mathbb{R}_+)$ and
$\widetilde{q}\in C(\mathbb{R}_+,\mathbb{R}_+)$ such that:
(1) $\widetilde{p}$ is concave,
$\widetilde{p}(0)=\widetilde{q}(0)=0$;
(2) There are two constants $r_2>0$ and $\mu_2>1$ such that
 \begin{gather*}
   f(t,x)\geq \widetilde{p}(x_3), g(t,x)\geq \widetilde{q}(x_1),
   \forall(t,x)\in[0,1]\times I_{r_2}^4,
   \\
\widetilde{p}(\widetilde{q}(t))\geq \frac{\pi^4\mu_2}{16}t, \forall t\in[0,r_2]
\end{gather*}

\item[(H6)] There are nonnegative constants
$a_{i}>0$, $b_{i}\geq0$, $c_{i}>0$, $d_{i}\geq 0$ $(i=3,4)$ and $c>0$ such that
\[
    \begin{pmatrix}
      f(t,x) \\
      g(t,x)
    \end{pmatrix}
\leq
    \begin{pmatrix}
      a_3 & 2b_3 & c_3 &2d_3 \\
      a_4 & 2b_4 & c_4 & 2d_4
    \end{pmatrix}
 \begin{pmatrix}
         x_1\\
         x_2 \\
         x_3 \\
         x_4
       \end{pmatrix}
+ \begin{pmatrix}
                 c \\
                 c
               \end{pmatrix}
\]
for all $(t,x)\in[0,1]\times\mathbb{R}_+^4$ and the matrix
\[
B_2:=\begin{pmatrix}
\lambda(a_3,b_3)-1,&-\beta(c_3,d_3,a_3,b_3,c_4,d_4)
\\ -\beta(a_4,b_4,c_4,d_4,a_3,b_3),&
\lambda(c_4,d_4)-1\end{pmatrix}
\]
 is an $\mathbb{R}_+^2$-monotone matrix,
 where the entries $\beta(c_3,d_3,a_3,b_3,c_4,d_4)$ and 
$\beta(a_4,b_4,c_4,d_4,a_3,b_3)$ are defined as in Lemma \ref{lem2.2}.

\item[(H7)] $f(t,x)$ and $g(t,x)$ are increasing in $x\in \mathbb{R}_+^4$,
and there is a constant $\omega>0$ such that
\[
  \int_0^1f(s,\omega,\omega,\omega,\omega)ds<\omega, \quad
  \int_0^1g(s,\omega,\omega,\omega,\omega)ds<\omega.
\]
\end{itemize}

\begin{remark}[{\cite[p.113]{Berman}}] \label{rmk1} \rm
 A real square matrix
$M$ is $\mathbb{R}_+^2$-monotone if and only if $M$ is nonsingular
and $M^{-1}$ is nonnegative.
\end{remark}

\begin{remark} \label{rmk2} \rm
Let $l_{ij}(i,j=1,2)$ be four nonnegative
 constants. Then it is easy to see that the matrix
 $D:=\begin{pmatrix}l_{11}-1&-l_{12} \\ -l_{21}&
l_{22}-1\end{pmatrix}$ is an $\mathbb{R}_+^2$-monotone matrix if
and only if $l_{11}>1$, $l_{22}>1,\det D=(l_{11}-1)(l_{22}-1)-l_{12}l_{21}>0$.
\end{remark}

\begin{remark} \label{rmk3}\rm
 $f(t,x)$ is said to be increasing in $x$ if \[f(t,x)\leq f(t,y)\]
 holds for every pair $x,y\in \mathbb {R}^{4}_+$ with $x\leq y$ for all
$t\in[0,1]$, where the partial ordering
 $\leq$ in $\mathbb{R}^{4}_+$ is understood componentwise.
\end{remark}

We adopt the convention in the sequel that $\widehat{c}_1,\widehat{c}_2,\dots$ stand
for different positive constants and
 $\Omega_\rho:=\{v\in E:\|v\|<\rho\}$ for $\rho>0$.

\begin{theorem} \label{thm3.1}
If {\rm (H1)--(H4)} hold, then
\eqref{second-order} has at least one positive solution.
\end{theorem}

\begin{proof} By (H2), we obtain
\begin{equation}\label{Thm31}
  A_1(u,v)(t)\geq\int_0^1k(t,s)p(v(s))ds-\widehat{c}_1,\quad
  A_2(u,v)(t)\geq\int_0^1k(t,s)q(u(s))ds-\widehat{c}_1
\end{equation}
for all $(u,v)\in P^2$, $t\in [0,1]$.
 We claim that the set
\[
 \mathcal{M}_1:=\{(u,v)\in P^2:
(u,v)=A(u,v)+\lambda(\sigma,\sigma), \lambda\geq0\}
\]
is bounded, where $\sigma(t):=te^{-t}$.  Indeed, if
$(u_{0},v_{0})\in\mathcal{M}_{1}$, then there exist a constant
$\lambda_{0}\geq0$ such that
$(u_{0},v_{0})=A(u_{0},v_{0})+\lambda_{0}(\sigma,\sigma)$, which can
be written in the form
\begin{gather*}
u_0(t)=\int_0^1k(t,s)f(s,u_0(s),u'_0(s),v_0(s),v_0'(s))ds+\lambda_0\sigma(t),\\
v_0(t)=\int_0^1k(t,s)g(s,u_0(s),u'_0(s),v_0(s),v_0'(s))ds+\lambda_0 \sigma(t).
\end{gather*}
By (H2) and  \eqref{Thm31}, we have
\begin{equation}\label{theorem1-1}
u_{0}(t)\geq\int_0^1k(t,s)p(v_{0}(s))ds-\widehat{c}_1,v_{0}(t)
\geq\int_0^1k(t,s)q(u_{0}(s))ds-\widehat{c}_1
\end{equation}
for all $t\in [0,1]$. The nonnegativity and  concavity of  $p$ imply
 $\lim_{y\to\infty}p(y)/y\geq 0$.
We also note $\max_{(t,s)\in[0,1]\times[0,1]}k(t,s)=1$. Now
Lemma \ref{lem2.6} and Jensen's inequality imply
\begin{equation}\label{theorem1-3}
p(v_{0}(t))\geq
p(v_{0}(t)+\widehat{c}_1)-p(\widehat{c}_1)\geq\int_0^1k(t,s)p(q(u_{0}(s)))ds
-p(\widehat{c}_1).
\end{equation}
This, together with  \eqref{theorem1-1} and (H2), implies
\begin{equation}\label{theorem1-4}
\begin{aligned}
u_{0}(t)&\geq\int_0^1k(t,s)
\Big[\int_0^1k(s,\tau)p(q(u_{0}(\tau)))d\tau-p(\widehat{c}_1)\Big]ds-\widehat{c}_1\\
&\geq\int_0^1\int_0^1k(t,s)k(s,\tau)p(q(u_{0}(\tau)))\,ds\,d\tau-\widehat{c}_2\\
&\geq\int_0^1\int_0^1k(t,s)k(s,\tau)\Big[\frac{\pi^4\mu_1}{16}u_{0}(\tau)-c\Big]ds\,
 d\tau-\widehat{c}_2\\
&\geq\frac{\pi^4\mu_1}{16}\int_0^1\int_0^1k(t,s)k(s,\tau)u_{0}(\tau)\,ds\,d\tau
-\widehat{c}_3.
\end{aligned}
\end{equation}
Multiply   both sides of the last inequality by
$\varphi(t):=\sin(\pi t/2)$ and integrate over $[0,1]$ and use
\eqref{varphi} twice to obtain
\begin{equation}\label{theorem1-5}
\begin{aligned}
\int_0^1\varphi(t)u_{0}(t)dt
&\geq\frac{\pi^4\mu_1}{16}\int_0^1\int_0^1\int_0^1\varphi(t)k(t,s)k(s,\tau)u_{0}(\tau)
dt\,ds\,d\tau-\frac{2\widehat{c}_3}{\pi}\\
&=\mu_1\int_0^1\varphi(t)u_{0}(t)dt-\frac{2\widehat{c}_3}{\pi},
\end{aligned}
\end{equation}
so that
\begin{equation}\label{theorem1-6}
  \int_0^1\varphi(t)u_{0}(t)dt\leq\frac{2\widehat{c}_3}{\pi(\mu_1-1)}.
\end{equation}
By Lemma \ref{lem2.1}, we have
\begin{equation}\label{theorem1-7}
\| u_{0}\|_{0}=u_0(1)\leq\frac{2\widehat{c}_3}{\pi\varrho(\varphi)(\mu_1-1)}
=\frac{\widehat{c}_3\pi}{2(\mu_1-1)}.
\end{equation}
Multiply  the first inequality of \eqref{theorem1-1} by
$\varphi(t)$,   integrate over $[0,1]$  and use \eqref{varphi} to obtain
\[
 \int_0^1u_0(t)\varphi(t)dt\geq\frac{4}{\pi^2}\int_0^1p(v_{0}(t))\varphi(t)dt
-\frac{2}{\pi}\widehat{c}_1.
\]
This, along with \eqref{theorem1-6}, implies
\begin{equation}\label{theorem1-8}
  \int_0^1p(v_{0}(t))\varphi(t)dt\leq\frac{\pi^2}{4}
\Big(\frac{2\widehat{c}_1}{\pi}+\int_0^1u_{0}(t)\varphi(t)
  dt\Big)
\leq  \frac{\widehat{c}_1\pi}{2}+\frac{\widehat{c}_3\pi}{2(\mu_1-1)}.
\end{equation}
By Lemma \ref{lem2.1}, we have
\begin{equation}\label{theorem1-9}
\begin{aligned}
  \|v_0\|_0=v_0(1)
&\leq\frac{1}{\varrho(\varphi)}  \int_0^1v_{0}(t)\varphi(t)dt\\
&=\frac{\|v_{0}\|_0}{\varrho(\varphi) p(\|v_0\|_0)}
 \int_0^1\varphi(t)\frac{v_{0}(t)}{\|v_0\|_0}p(\|v_{0}\|_0)dt\\
&\leq\frac{\|v_{0}\|_0}{\varrho(\varphi) p(\|v_{0}\|_0)}
 \int_0^1\varphi(t)p(v_{0}(t))dt,
  \end{aligned}
\end{equation}
so that
\[
p(\|v_{0}\|_0)\leq\frac{1}{\varrho(\varphi) }\int_0^1\varphi(t)p(v_{0}(t))dt
\leq\frac{\widehat{c}_1\pi^3}{8}+\frac{\widehat{c}_3\pi^4}{16(\mu_1-1)}.
\]
 (H2) implies that $p$ is strictly increasing and
$\lim_{x\to\infty}p(x)=\infty$ (see Lemma \ref{lem2.6}). Consequently, there
exists $\widehat{c}_4>0$ such that
 \[
\|v_{0}\|_0\leq \widehat{c}_4.
\]
 Let $N:=\max \{\frac{\widehat{c}_3\pi}{2(\mu_1-1)}, \widehat{c}_4\}$. Then
\begin{equation}\label{theorem1-10}
\|u\|_0\leq N,\quad \|v\|_0\leq N, \quad \forall (u,v)\in\mathcal{M}_1.
\end{equation}
This establishes  the a priori bound of $\mathcal{M}_1$ for
$\|(u,v)\|_0$. Now it remains to derive the a priori bound of
$\mathcal{M}_1$ for $\|(u',v')\|_0$. To this end, we let
\[
\Lambda:=\{\mu\geq 0:  \text{there exists $(u,v)\in P^2$
\ such that } (u,v)=A(u,v)+\mu(\sigma,\sigma)\}.
\]
 Now
\eqref{theorem1-10} imply that $\mu_{0}:=\sup\Lambda<\infty$. By
(H3), there are two functions $\Phi_N, \Psi_N\in C(\mathbb{R}_+,\mathbb{R}_+)$
such that
\begin{gather*}
f(t,u(t),u'(t),v(t),v'(t))\leq\Phi_N(u'(t)+v'(t)),\\
g(t,u(t),u'(t),v(t),v'(t))\leq\Psi_N(u'(t)+v'(t))
\end{gather*}
for all $(u,v)\in\mathcal {M}_{1}$, $t\in [0,1]$. Hence,  for all
$(u,v)\in\mathcal{M}_1$ and for some $\mu\geq 0$,  we have
\begin{align*}
 -u''(t)
&=f(t,u(t),u'(t),v(t),v'(t))+\mu(2-t)e^{-t}\\
&\leq\Phi_N(u'(t)+v'(t))+\mu(2-t)e^{-t}\\
&\leq\Phi_N(u'(t)+v'(t))+2\mu_0,
\end{align*}
\begin{align*}
-v''(t)&=g(t,u(t),u'(t),v(t),v'(t))+\mu(2-t)e^{-t}\\
&\leq\Psi_N(u'(t)+v'(t))+\mu(2-t)e^{-t}\\
&\leq\Psi_N(u'(t)+v'(t))+2\mu_0,
\end{align*}
so that
\begin{align*}
  -(u''(t)+v''(t))(u'(t)+v'(t))
&\leq(u'(t)   +v'(t))(\Phi_N(u'(t)+v'(t))\\
&\quad +\Psi_N(u'(t)+v'(t))+4\mu_0).
\end{align*}
This implies
\[
  \int_0^{u'(0)+v'(0)}\frac{\tau d\tau}{\Phi_N(\tau)+\Psi_N(\tau)+4\mu_0}
  \leq\int_0^1(u'(t)+v'(t))dt=u(1)+v(1)\leq 2N
\]
for all $(u,v)\in\mathcal {M}_{1}$. By (H3) again, there exists a
constant $N_1>0$ such that
\[
  \|u'+v'\|_0=u'(0)+v'(0)\leq N_1,\quad
  \forall (u,v)\in\mathcal {M}_{1}.
\]
This  establishes the a priori bound of $\mathcal{M}_1$ for
$\|(u',v')\|_0$ and, in turn,  implies that
$\mathcal{M}_1$ is bounded(notice that we have achieved the  a
priori bound of $\mathcal{M}_1$ for $\|(u,v)\|_0$ in
\eqref{theorem1-10}). Taking
$R>\max\{\sup\{\|(u,v)\|:(u,v)\in\mathcal{M}_1\},r\}$, we have
\[
 (u,v)\neq A(u,v)+\lambda(\sigma,\sigma), \quad \forall
 (u,v)\in\partial\Omega_R\cap P^2, \lambda\geq0.
\]
Now Lemma \ref{lem2.4} yields
\begin{equation}\label{theorem2}
  i(A,\Omega_R\cap P^2,P^2)=0.
\end{equation}
Let
\[
\mathcal{M}_2:=\{(u,v)\in\overline{\Omega}_r\cap
P^2:(u,v)=\lambda A(u,v), \lambda\in[0,1]\}.
\]
 Now we want to prove that $\mathcal{M}_2=\{0\}$. Indeed,
if $(u,v)\in\mathcal{M}_2$, then
$(u,v)\in P_0^2$ and $(u,v)=\lambda A(u,v)$ for some
$\lambda\in[0,1]$, written componentwise as
\begin{gather*}
 u(t)=\lambda\int_0^1k(t,s)f(s,u(s),u'(s),v(s),v'(s))ds,\\
 v(t)=\lambda\int_0^1k(t,s)g(s,u(s),u'(s),v(s),v'(s))ds,
\end{gather*}
which are equivalent to
\[
  -u''(t)=\lambda f(t,u(t),u'(t),v(t),v'(t)),
  -v''(t)=\lambda g(t,u(t),u'(t),v(t),v'(t)).
\]
By (H4), we have
\begin{gather*}
-u''(t)\leq  a_{1}u(t)+2b_{1}u'(t)+c_{1}v(t)+2d_{1}v'(t), \\
-v''(t)\leq  a_{2}u(t)+2b_{2}u'(t)+c_{2}v(t)+2d_{2}v'(t).
\end{gather*}
Multiply  the last two inequalities by $\varphi_{a_{1},b_{1}}(t)$
and $\varphi_{c_{2},d_{2}}(t)$ respectively and integrate over
$[0,1]$ and use Lemmas \ref{lem2.2} and \ref{lem2.3} to obtain
\begin{align*}
 &\lambda(a_{1},b_{1})\int_0^1(a_{1}u(t)+2b_{1}u'(t))\varphi_{a_{1},b_{1}}(t)dt\\
  &\leq\int_0^1(a_{1}u(t)+2b_{1}u'(t))\varphi_{a_{1},b_{1}}(t)dt+\int_0^1(c_{1}v(t)+2d_{1}v'(t))
  \varphi_{a_{1},b_{1}}(t)dt\\
  &\leq\int_0^1(a_{1}u(t)+2b_{1}u'(t))\varphi_{a_{1},b_{1}}(t)dt\\
&\quad  +\beta(c_{1},d_{1},a_{1},b_{1},c_{2},d_{2})\int_0^1(c_2v(t)
  +2d_2v'(t))\varphi_{c_2,d_2}(t)dt,
\end{align*}
\begin{align*}
&\lambda(c_2,d_2)\int_0^1(c_2v(t)+2d_2v'(t))\varphi_{c_2,d_2}(t)dt\\
&\leq\int_0^1(a_{2}u(t)+2b_{2}u'(t))\varphi_{c_2,d_2}(t)dt
  +\int_0^1(c_{2}v(t)+2d_{2}v'(t))   \varphi_{c_2,d_2}(t)dt\\
&\leq\beta(a_2,b_2,c_2,d_2,a_1,b_1)\int_0^1(a_1u(t)
  +2b_1u'(t))\varphi_{a_1,b_1}(t)dt\\
&\quad +\int_0^1(c_2v(t)+2d_2v'(t))\varphi_{c_2,d_2}(t)dt,
\end{align*}
which can be written in the form
\begin{align*}
&\begin{pmatrix}\lambda(a_1,b_1)-1&-\beta(c_1,d_1,a_1,b_1,c_2,d_2) \\
  -\beta(a_2,b_2,c_2,d_2,a_1,b_1)&
\lambda(c_2,d_2)-1\end{pmatrix}\\
&\cdot  \begin{pmatrix}
\int_0^1(a_1u(t)+2b_1u'(t))\varphi_{a_1,b_1}(t)dt\\
  \int_0^1(c_2v(t)+2d_2v'(t))\varphi_{c_2,d_2}(t)dt
  \end{pmatrix}
\\
&=B_1 \begin{pmatrix}
\int_0^1(a_1u(t)+2b_1u'(t))\varphi_{a_1,b_1}(t)dt
  \\ \int_0^1(c_2v(t)+2d_2v'(t))\varphi_{c_2,d_2}(t)dt
  \end{pmatrix}\leq\begin{pmatrix}0 \\ 0
  \end{pmatrix}.
\end{align*}
(H4) again implies
\[
 \begin{pmatrix}
\int_0^1(a_1u(t)+2b_1u'(t))\varphi_{a_1,b_1}(t)dt \\
\int_0^1(c_2v(t)+2d_2v'(t))\varphi_{c_2,d_2}(t)dt
  \end{pmatrix}
\leq B_1^{-1}\begin{pmatrix}0 \\0
  \end{pmatrix}=\begin{pmatrix}0 \\0
  \end{pmatrix}.
\]
Consequently,
\[
\int_0^1(a_1u(t)+2b_1u'(t))\varphi_{a_1,b_1}(t)dt
=\int_0^1(c_2v(t)+2d_2v'(t))\varphi_{c_2,d_2}(t)dt=0
\]
and  $u=v=0$, whence $\mathcal{M}_2= \{0\}$, as required. As a result
of this, we have
\[(u,v)\neq\lambda A(u,v), \quad \forall
(u,v)\in\partial\Omega_r\cap P^2, \lambda\in[0,1].
\]
Now Lemma \ref{lem2.5} yields
\[
  i(A,\Omega_r\cap P^2,P^2)=1.
\]
This together with \eqref{theorem2} implies
\[
  i(A,(\Omega_R\backslash\overline{\Omega}_r)\cap P^2,P^2)=0-1=-1.
\]
Therefore, $A$ has at least one fixed point $(u,v)$ on
$(\Omega_R\backslash\overline{\Omega}_r)\cap P^2$ and thus
  \eqref{second-order} has at least one positive solution.
This completes the proof.
\end{proof}

\begin{theorem} \label{thm3.2}
If {\rm (H1), (H5), (H6)} hold, then
\eqref{second-order} has at least one positive solution.
\end{theorem}


\begin{proof} By (H5), for all
$(u,v)\in\overline{\Omega}_{r_2}\cap P^2$, we have
\[
  A_1(u,v)(t)\geq\int_0^1k(t,s)\widetilde{p}(v(s))ds, \quad
A_2(u,v)(t)\geq\int_0^1k(t,s)\widetilde{q}(u(s))ds.
\]
Let
\[
  \mathcal{M}_3:=\{(u,v)\in\overline{\Omega}_{r_2}\cap P^2 :(u,v)
=A(u,v)+\lambda(\sigma,\sigma),\lambda\geq0\}.
\]
Now we want to prove that $\mathcal{M}_3\subset\{0\}$, where
$\sigma(t):=te^{-t}$. If
$(\widetilde{u},\widetilde{v})\in\mathcal{M}_3$, there exists
$\widetilde{\lambda}\geq0$  such that
$(\widetilde{u},\widetilde{v})=A(\widetilde{u},\widetilde{v})
+\widetilde{\lambda}(\sigma,\sigma)$,
which implies
\begin{equation}\label{theorem2-1}
  \widetilde{u}(t)\geq\int_0^1k(t,s)\widetilde{p}(\widetilde{v}(s))ds,
\widetilde{v}(t)\geq\int_0^1k(t,s)\widetilde{q}(\widetilde{u}(s))ds
\end{equation}
Note $\max_{(t,s)\in[0,1]\times[0,1]}k(t,s)=1$. By (H5) and Jensen's
inequality, we have
\begin{align*}
  \widetilde{u}(t)
&\geq\int_0^1k(t,s)\widetilde{p}(\widetilde{v}(s))ds\\
&\geq\int_0^1k(t,s)\widetilde{p}(\int_0^1k(s,\tau)\widetilde{q}(\widetilde{u}(\tau))d\tau)ds\\
&\geq\int_0^1\int_0^1k(t,s)k(s,\tau)\widetilde{p}(\widetilde{q}(\widetilde{u}(\tau)))d\tau ds\\
&\geq\int_0^1\int_0^1k(t,s)k(s,\tau)\widetilde{p}(\widetilde{q}(\widetilde{u}(\tau)))\,ds\,d\tau\\
&\geq\frac{\pi^4\mu_2}{16}\int_0^1\int_0^1k(t,s)k(s,\tau)\widetilde{u}(\tau)\,ds\,d\tau\,.
 \end{align*}
Multiply   both sides of the above inequality   by
$\varphi(t):=\sin\frac{\pi}{2}t$ and integrate over $[0,1]$ and use
\eqref{varphi} to obtain
\[
  \int_0^1\widetilde{u}(t)\varphi(t)dt
\geq\mu_2\int_0^1\widetilde{u}(t)\varphi(t)dt,
\]
so that  $\int_0^1\widetilde{u}(t)\varphi(t)dt=0$, and whence
$\widetilde{u}(t)\equiv 0$. This, together with \eqref{theorem2-1},
yields $\widetilde{p}(\widetilde{v}(t))\equiv 0$, and, in
particular,
\[
\widetilde{p}(\|\widetilde{v}\|_0)=\widetilde{p}(\widetilde{v}(1))= 0.
\]
Note that (H5) implies that $\widetilde{p}$ is strictly increasing on
$[0, \varepsilon]$ for sufficiently small $\varepsilon>0$ (see Lemma \ref{lem2.6})
and thus $\widetilde{v}(1)=0$. Hence
$\widetilde{u}=\widetilde{v}=0$, and $\mathcal{M}_3\subset\{0\}$, as
required. As a result of this, we have
\[
  (u,v)\neq A(u,v)+\lambda(\varphi,\varphi), \forall
  (u,v)\in\partial\Omega_{r_2}\cap  P^2, \lambda\geq0.
\]
Now Lemma \ref{lem2.4} yields
\begin{equation}\label{theorem6}
i(A,\Omega_{r_2}\cap  P^2,P^2)=0.
\end{equation}
 Let
\[
  \mathcal{M}_{4}:=\{(u,v)\in P^2:(u,v)=\lambda A(u,v),
  \lambda\in[0,1]\}.
\]
We now assert that $\mathcal{M}_4$ is bounded. Indeed,  if
$(u,v)\in\mathcal{M}_{4}$, then  $(u,v)\in P_0^2$ and $(u,v)=\lambda
A(u,v)$ for some  $\lambda\in[0,1]$, which can be written
componentwise as
\begin{gather*}
  u(t)=\lambda\int_0^1k(t,s)f(s,u(s),u'(s),v(s),v'(s))d  s,\\
  v(t)=\lambda\int_0^1k(t,s)g(s,u(s),u'(s),v(s),v'(s))ds.
\end{gather*}
Differentiate the last equations twice to obtain
\[
  -u''(t)=\lambda f(t,u(t),u'(t),v(t),v'(t)), \quad
  -v''(t)=\lambda g(t,u(t),u'(t),v(t),v'(t)),
\]
for $t\in [0,1]$.
By (H6), we have
\begin{gather}\label{theorem3}
-u''(t)\leq a_{3}u(t)+2b_{3}u'(t)+c_{3}v(t)+2d_{3}v'(t)+\widetilde{c},\\
\label{theorem4}
-v''(t)\leq
a_{4}u(t)+2b_{4}u'(t)+c_{4}v(t)+2d_{4}v'(t)+\widetilde{c}
\end{gather}
Multiply  the last two inequalities by $\varphi_{a_3,b_3}(t)$ and
$\varphi_{c_4,d_4}(t)$ respectively and integrate over $[0,1]$, and
use Lemmas \ref{lem2.2} and  \ref{lem2.3} to obtain
\[\begin{aligned}
 \lambda(a_3,b_3)&\int_0^1(a_3u(t)+2b_3u'(t))\varphi_{a_3,b_3}(t)dt\\
  &\leq\int_0^1(a_3u(t)+2b_3u'(t))\varphi_{a_3,b_3}(t)dt+
  \int_0^1(c_3v(t)+2d_3v'(t))\varphi_{a_3,b_3}(t)dt+\widetilde{c}\\
  &\leq\int_0^1(a_3u(t)+2b_3u'(t))\varphi_{a_3,b_3}(t)dt\\
  &+
  \beta(c_3,d_3,a_3,b_3,c_4,d_4)\int_0^1(c_4v(t)+2d_4v'(t))\varphi_{c_4,d_4}(t)dt+\widetilde{c},
\end{aligned}\]
and
\begin{align*}
 &\lambda(c_4,d_4)\int_0^1(c_4v(t)+2d_4v'(t))\varphi_{c_4,d_4}(t)dt\\
  &\leq\int_0^1(a_4u(t)+2b_4u'(t))\varphi_{c_4,d_4}(t)dt
  +\int_0^1(c_4v(t)+2d_4v'(t))\varphi_{c_4,d_4}(t)dt+\widetilde{c}\\
  &\leq\beta(a_4,b_4,c_4,d_4,a_3,b_3)\int_0^1(a_3u(t)+2b_3u'(t))\varphi_{a_3,b_3}(t)dt\\
  &\quad +\int_0^1(c_4v(t)+2d_4v'(t))\varphi_{c_4,d_4}(t)dt+\widetilde{c},
\end{align*}
which can be written in the form
\begin{align*}
&\begin{pmatrix}
\lambda(a_3,b_3)-1 &-\beta(c_3,d_3,a_3,b_3,c_4,d_4) \\
  -\beta(a_4,b_4,c_4,d_4,a_3,b_3)&
\lambda(c_4,d_4)-1\end{pmatrix}\\
&\cdot \begin{pmatrix}\int_0^1(a_3u(t)+2b_3u'(t))\varphi_{a_3,b_3}(t)dt \\
\int_0^1(c_4v(t)+2d_4v'(t))\varphi_{c_4,d_4}(t)dt
  \end{pmatrix}\\
&=B_2 \begin{pmatrix}\int_0^1(a_3u(t)+2b_3u'(t))\varphi_{a_3,b_3}(t)dt \\
\int_0^1(c_4v(t)+2d_4v'(t))\varphi_{c_4,d_4}(t)dt
  \end{pmatrix}
\leq\begin{pmatrix}\widetilde{c} \\ \widetilde{c}
  \end{pmatrix}.
\end{align*}
(H6) again implies
\[
\begin{pmatrix}
\int_0^1(a_3u(t)+2b_3u'(t))\varphi_{a_3,b_3}(t)dt \\
\int_0^1(c_4v(t)+2d_4v'(t))\varphi_{c_4,d_4}(t)dt \end{pmatrix}
\leq B_2^{-1}\begin{pmatrix}\widetilde{c} \\ \widetilde{c}
  \end{pmatrix}:=\begin{pmatrix}\widetilde{c}_1 \\ \widetilde{c}_2
  \end{pmatrix}.
\]
 Let $\widetilde{c}_3:=\max\{\widetilde{c}_1,\widetilde{c}_2\}>0$. Then  we have
\[
  \int_0^1(a_3u(t)+2b_3u'(t))\varphi_{a_3,b_3}(t)dt\leq \widetilde{c}_3,
  \int_0^1(c_4v(t)+2d_4v'(t))\varphi_{c_4,d_4}(t)dt\leq \widetilde{c}_3,
\]
for all $ (u,v)\in\mathcal{M}_4$.
By Lemma \ref{lem2.1}, we obtain
\[
  \|u\|_0=u(1)\leq\frac{\widetilde{c}_3}{a_3\varrho(\varphi_{a_3,b_3})}, \quad
  \|v\|_0=v(1)\leq\frac{\widetilde{c}_3}{c_4\varrho(\varphi_{c_4,d_4})},
\]
for all $(u,v)\in\mathcal{M}_4$. Let
$\widetilde{N}=\max\{\frac{\widetilde{c}_3}{a_3\varrho(\varphi_{a_3,b_3})},
\frac{\widetilde{c}_3}{c_4\varrho(\varphi_{c_4,d_4})}\}>0$. Then  by
\eqref{theorem3} and \eqref{theorem4},  we have
\begin{gather*}
 -u''(t)\leq  (a_3+c_3)\widetilde{N}+2b_3u'(t)+2d_3v'(t)+\widetilde{c}, \\
 -v''(t)\leq  (a_4+c_4)\widetilde{N}+2b_4u'(t)+2d_4v'(t)+\widetilde{c},
\end{gather*}
for all $(u,v)\in\mathcal{M}_4$. Adding the above inequalities yields
\[
-u''(t)-v''(t)\leq
(a_3+a_4+c_3+c_4)\widetilde{N}
+2(b_3+b_4)u'(t)+2(d_3+d_4)v'(t)+2\widetilde{c}.
\]
 Let
\[
\widetilde{N}_2:=(a_3+a_4+c_3+c_4)\widetilde{N}+2\widetilde{c},
\widetilde{L}:=2(b_3+b_4+d_3+d_4)+1.
\] 
Noticing $u'(1)=v'(1)=0$, we obtain
\[
  u'(t)+v'(t)\leq\frac{\widetilde{N}_2}{\widetilde{L}}(e^{\widetilde{L}
-\widetilde{L}t}-1),
\]
so that
\[
  \|u'+v'\|_0=u'(0)+v'(0)\leq
  \frac{\widetilde{N}_2}{\widetilde{L}}(e^{\widetilde{L}}-1).
\]
This proves the boundedness of $\mathcal{M}_4$, as asserted. Taking
$R>\max\{\sup\{\|(u,v)\|:(u,v)\in\mathcal{M}_4\},r_2\}$, we have
\[
  (u,v)\neq\lambda A(u,v), \quad \forall(u,v)\in\partial\Omega_R\cap(P\times
  P),\; \lambda\in[0,1].
\]
Now Lemma \ref{lem2.5} yields
\begin{equation}\label{theorem5}
  i(A,\Omega_R\cap P^2,P^2)=1.
\end{equation}
Combining  \eqref{theorem6} and \eqref{theorem5} gives
\[
i(A,(\Omega_R\backslash\overline{\Omega}_{r_2})\cap  P^2,P^2)=1.
\]
Hence $A$ has at least one fixed point on
$(\Omega_R\backslash\overline{\Omega}_{r_2})\cap  P^2$. Thus
\eqref{second-order} has at least one positive solution. This
completes the proof.
\end{proof}

\begin{theorem} \label{thm3.3} 
If {\rm (H1)--(H3), (H5), (H7)} hold, then
\eqref{second-order} has at least two
positive solutions.
\end{theorem}

\begin{proof} By (H7), we have
\[
 f(t,x)\leq f(t,\omega,\omega,\omega,\omega),\quad
 g(t,x)\leq g(t,\omega,\omega,\omega,\omega),
\]
for all $t\in[0,1]$ and all $x\in I_\omega^4$. Consequently, we
have for all $(u,v)\in
\partial\Omega_{\omega}\cap  P^2$,
\begin{align*}
\|A_1(u,v)\|_0=A_1(u,v)(1)
&=\int_0^1sf(s,u(s),u'(s),v(s),v'(s))ds\\
&\leq\int_0^1f(s,u(s),u'(s),v(s),v'(s))ds\\
&\leq\int_0^1f(s,\omega,\omega,\omega,\omega)ds\\
&<\omega=\|(u,v)\|,
\end{align*}
\begin{align*}
\|A_2(u,v)\|_0=A_2(u,v)(1)
&=\int_0^1sg(s,u(s),u'(s),v(s),v'(s))ds\\
&\leq\int_0^1g(s,u(s),u'(s),v(s),v'(s))ds\\
&\leq\int_0^1g(s,\omega,\omega,\omega,\omega)ds\\
&<\omega=\|(u,v)\|,
\end{align*}
\begin{align*}
\|(A_1(u,v))'\|_0
&=(A_1(u,v))'(0)=\int_0^1f(s,u(s),u'(s),v(s),v'(s))ds\\
&\leq\int_0^1f(s,\omega,\omega,\omega,\omega)ds\\
&<\omega=\|(u,v)\|,
\end{align*}
and
\begin{align*}
\|(A_2(u,v))'\|_0
&=(A_2(u,v))'(0)=\int_0^1g(s,u(s),u'(s),v(s),v'(s))ds\\
&\leq\int_0^1g(s,\omega,\omega,\omega,\omega)ds\\
&<\omega=\|(u,v)\|.
\end{align*}
The preceding inequalities imply
\[
\|A(u,v)\|=\|(A_1(u,v),A_2(u,v))\|<\omega=\|(u,v)\|;
\]
 thus
\[
  (u,v)\neq\lambda A(u,v), \quad \forall (u,v)\in\partial\Omega_{\omega}\cap  P^2,\;
 0\leq\lambda\leq1.
\]
Now Lemma \ref{lem2.5} yields
\begin{equation}\label{theorem7}
  i(A,\Omega_{\omega}\cap  P^2,P^2)=1.
\end{equation}
By (H2), (H3) and (H5), we find  that \eqref{theorem2} and
\eqref{theorem6} hold. Note that we can choose $R>\omega>r_2$ in
\eqref{theorem2} and \eqref{theorem6} (see the proofs of Theorems
\ref{thm3.1} and \ref{thm3.2}). Combining \eqref{theorem2}, \eqref{theorem6} and
\eqref{theorem7}, we obtain
\begin{gather*}
i(A,(\Omega_{R}\backslash\overline{\Omega}_{\omega})\cap
P^2,P^2)=0-1=-1, \\
i(A,(\Omega_{\omega}\backslash\overline{\Omega}_{r_2})\cap
P^2,P^2)=1-0=1.
\end{gather*}
Therefore, $A$ has at least two fixed points, with one on
$(\Omega_{R}\backslash\overline{\Omega}_{\omega})\cap  P^2$ and the
other on $(\Omega_{\omega}\backslash\overline{\Omega}_{r_2})\cap
P^2$. Hence \eqref{second-order} has at least two positive
solutions.
\end{proof}

\section{Examples}

In this section we present four  examples to   illustrate our main
results.

\begin{example} \label{examp4.1} \rm
 Let    $(a_{ij})_{2\times4}$ be a positive
matrix with  $1<\alpha_i\leq 2(i=1,2)$  and
\[
f(t,x):=\Big(\sum_{j=1}^{4} a_{1j}x_j\Big)^{\alpha_1}, \quad
g(t,x):=\Big(\sum_{j=1}^{4} a_{2j}x_j\Big)^{\alpha_2},\quad
x\in\mathbb{R}_+^{4},\; t\in [0,1].
\] 
Then (H1)-(H4) holds with both $f$
and $g$  superlinear. By Theorem \ref{thm3.1}, Equaton \eqref{second-order} has at
least one positive solution. It suffices to verify (H2)-(H4).

(1) Let $p(y):=y$ and $q(y):=a_{21} y^{\alpha_2}$. Then $p$ is concave and
$p(q(y))/y\to \infty(y\to \infty)$. It is easy to see that there
exists $c>0$ such that
\[
f(t,x)\geq p(x_3)-c, g(t,x)\geq q(x_1)-c
\]
for all $x\in\mathbb{R}_+^4$ and $t\in [0,1]$. This implies that (H2)
holds true.

(2) Assumption (H3) holds with 
 \[
\Phi_N(t):=\big((a_{12}+a_{14})t+2N\big)^{\alpha_1}, \quad
\Psi_N(t):=\big((a_{22}+a_{24})t+2N\big)^{\alpha_2}
\] 
for every  $N>0$.


(3) Note that there is $r>0$ such that
\[
 \begin{pmatrix}
      f(t,x) \\
      g(t,x) 
    \end{pmatrix}
\leq    \frac{1}{3}
    \begin{pmatrix}
      1& 2 & 1 & 2 \\
      1& 2 & 1 &2 
    \end{pmatrix}
  \begin{pmatrix}
    x_1 \\
    x_2 \\
    x_3 \\
    x_4 
  \end{pmatrix}
\]
for all $x\in I_r^4, t\in [0,1]$. Let $a:=\frac{1}{3}$. Then
\[
\lambda (a,a)=\frac{1}{a}=3, \beta(a,a,a,a,a,a)=1.
\]
Obviously,  the matrix 
\[ B_1: = \begin{pmatrix}
  \lambda (a,a)-1 & -\beta(a,a,a,a,a,a) \\
  -\beta(a,a,a,a,a,a) & \lambda (a,a)-1 
\end{pmatrix}
=    \begin{pmatrix}
      2 & -1 \\
      -1 & 2 
    \end{pmatrix}
\]
 is $\mathbb{R}_2^+$-monotone. Therefore (H4) holds with
$a_i=b_i=c_i=d_i=a(i=1,2)$ and $B_1$ as defined above.
\end{example}



\begin{example} \label{examp4.2}\rm
 Let    $(b_{ij})_{2\times 4}$ be a positive
matrix  with $0<\alpha_i<1(i=3,4)$  and
\[
f(t,x):=\Big(\sum_{j=1}^{4} b_{1j}x_j\Big)^{\alpha_3},\quad
g(t,x):=\Big(\sum_{j=1}^{4}b_{2j}x_j\Big)^{\alpha_4}, \quad
x\in\mathbb{R}_+^{4},\; t\in [0,1].
\] 
 Now  (H1), (H5) and (H6) hold  with both $f$
and $g$  sublinear. By Theorem \ref{thm3.2}, \eqref{second-order} has at
least one positive solution.
\end{example}

\begin{example} \label{examp4.3} \rm
Let    $(c_{ij})_{2\times 4}$ be a positive matrix  and
\[
f(t,x):=\frac{\sum_{j=1}^{4}c_{1j}x_j^{\frac{5}{3}}}{\sum_{j=1}^{4}x_j+1},\quad
g(t,x):=\frac{\sum_{j=1}^{4}c_{2j}x_j^3}{\sum_{j=1}^{4}x_j+1},\quad
x\in\mathbb{R}_+^{4}, \; t\in [0,1].
\]
 Now (H1)--(H4) hold with $f$  sublinear and $g$ superlinear  at
$\infty$. By Theorem \ref{thm3.1}, Equation \eqref{second-order} has at least one
positive solution.
\end{example}

\begin{example} \label{examp4.4} \rm
Let $(a_{ij})_{2\times 4}$, $(b_{ij})_{2\times
4}$  be two positive matrices, with $1<\beta_i\leq 2$,
$0<\gamma_i<1(i=1,2)$ and
\[
\Big(\sum_{j=1}^{4} a_{1j}\Big)^{\beta_1}
+\Big(\sum_{j=1}^{4}b_{1j}\Big)^{\gamma_1}<1, \quad
\Big(\sum_{j=1}^{4}a_{2j}\Big)^{\beta_2}
+\Big(\sum_{j=1}^{4} b_{2j}\Big)^{\gamma_2}<1.
\]
Let
\begin{gather*}
f(t,x):=\Big(\sum_{j=1}^{4} a_{1j}x_j\Big)^{\beta_1}
+\Big(\sum_{j=1}^{4} b_{1j}x_j\Big)^{\gamma_1}, \\
g(t,x):=\Big(\sum_{j=1}^{4}a_{2j}x_j\Big)^{\beta_2}
+\Big(\sum_{j=1}^{4} b_{2j}x_j\Big)^{\gamma_2} 
\end{gather*}
for $x\in\mathbb{R}_+^4$, 
$t\in [0,1]$.  Now (H1)--(H3), (H5) and (H7) hold  with both $f$ and $g$
superlinear at $\infty$ and sublinear at $0$. By Theorem \ref{thm3.3},
\eqref{second-order} has at least two positive solutions.
\end{example}

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\end{document}