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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 52, pp. 1--15.\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-2012/52\hfil Positive solutions for a system]
{Positive solutions for a system of higher order boundary-value problems
 involving all derivatives of odd orders}

\author[K. Wang, Z. Yang\hfil EJDE-2012/52\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 October 24, 2011. Published March 30, 2012.}
\subjclass[2000]{34B18, 45G15, 45M20, 47H07, 47H11}
\keywords{Systenm of higher order boundary value problem; positive solution; 
\hfill\break\indent  nonnegative matrix; fixed point index}

\begin{abstract}
 In this article we study the existence of positive
 solutions for the system of higher order boundary-value problems
 involving all derivatives of odd orders
 \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*}
 Here $f,g\in C([0,1]\times\mathbb{R}_+^{m+n+2},\mathbb{R}_+)$
 $(\mathbb{R}_+:=[0,+\infty))$.
 Our hypotheses imposed on the nonlinearities $f$ and $g$ are
 formulated in terms of two linear functions $h_1(x)$ and $h_2(y)$.
 We use fixed point index theory to establish our main results based
 on a priori estimates of positive solutions achieved by utilizing
 nonnegative 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 higher order boundary-value
problems:
\begin{equation}\label{higher-order}
\begin{gathered}
(-1)^mw^{(2m)}=f(t, w, w',\dots, (-1)^{m-1}w^{(2m-1)}, z,
z',\dots, (-1)^{n-1}z^{(2n-1)}),\\
(-1)^nz^{(2n)}=g(t, w, w',\dots, (-1)^{m-1}w^{(2m-1)}, z,
z',\dots, (-1)^{n-1}z^{(2n-1)}),\\
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{gathered}
\end{equation}
where $m,n\geq 2, f\in C([0,1]\times \mathbb{R}_+^{m+n+2},\mathbb{R}_+)$ and $g\in C([0,1]\times \mathbb{R}_+^{m+n+2},\mathbb{R}_+)$
$(\mathbb{R}_+:=[0,+\infty))$. By a positive solution of
\eqref{higher-order}, we mean a pair of functions $(w,z)\in
C^{2m}[0,1]\times C^{2n}[0,1]$ that solve \eqref{higher-order} and
satisfy $w(t)\geq 0$, $z(t)\geq 0$ for all $t\in[0,1]$,
with at least one of them positive on $(0,1]$.

The so-called Lidstone problem
\begin{equation}\label{Lidstone}
\begin{gathered}
(-1)^nu^{(2n)}=f(t, u, -u'',\dots, (-1)^{n-1}u^{(2n-2)}),\\
 u^{(2i)}(0)=u^{(2i)}(1)=0, \quad (i=0,1,\dots, n-1),
\end{gathered}
\end{equation}
has been extensively studied in recent years;
see \cite{Yao,YuhongMa,YuanmingWang1,YuanmingWang2,Wei1,Wei2} and the references 
cited therein.
The existence of positive solutions for systems of nonlinear differential equations 
have been studied by many authors; see for instance, 
\cite{Dunninger1,Henderson1,Liu1,Cao1}, to cite a few. In
\cite{Yang2}, the author studied the existence of positive solutions
of the system
\begin{gather*}
(-1)^mu^{(2m)}=f_1(t, u,-u'',\dots,
(-1)^{m-1}u^{(2m-2)}, v,-v'',\dots,
(-1)^{n-1}v^{(2n-2)}),\\
(-1)^nv^{(2n)}=f_2(t, u,-u'',\dots,
(-1)^{m-1}u^{(2m-2)}, v,-v'',\dots,
(-1)^{n-1}v^{(2n-2)}),\\
\alpha_0u^{(2i)}(0)-\beta_0u^{(2i+1)}(0)
=\alpha_1u^{(2i)}(1)+\beta_1u^{(2i+1)}(1)=0\quad (i=0,1,\dots, m-1),\\
\alpha_0v^{(2j)}(0)-\beta_0v^{(2j+1)}(0)
=\alpha_1v^{(2j)}(1)+\beta_1v^{(2j+1)}(1)=0\quad (j=0,1,\dots,n-1),
\end{gather*}
where $m,n\geq 1$ and $f_1,f_2\in C([0,1]\times \mathbb{R}_+^{m+n},\mathbb{R}_+)$. The main results obtained in \cite{Yang2}
are presented in terms of nonnegative matrices and the author used
the method of order reduction  to overcome the difficulty arising from 
high order derivatives. Furthermore, in \cite{Kang}, Kang et al.,
using the fixed point theorem of cone expansion and compression type due to 
Krasnosel'skill, established some simple criteria for the existence,
 multiplicity and nonexistence of positive solutions of the following 
systems of singular boundary value problems with integral boundary value conditions:
\begin{gather*}
(-1)^pu^{(2p)}=\lambda a_1(t)f(t,u,-u'',\dots,(-1)^{p-1}u^{2p-2},v,-v'',
 \dots,(-1)^{q-1}v^{2q-2}),\\
(-1)^qv^{(2q)}=\mu a_2(t)g(t,u,-u'',\dots,(-1)^{p-1}u^{2p-2},v,-v'',
 \dots,(-1)^{q-1}v^{2q-2}),\\
a_iu^{(2i)}(0)-b_iu^{(2i+1)}(0)=\int_0^1m_i(s)u^{(2i)}(s)ds,\quad 0\leq i\leq p-1,\\
c_iu^{(2i)}(1)-d_iu^{(2i+1)}(1)=\int_0^1n_i(s)u^{(2i)}(s)ds, \quad 0\leq i\leq p-1,\\
\alpha_jv^{(2j)}(0)-\beta_jv^{(2j+1)}(0)=\int_0^1\varphi_j(s)v^{(2j)}(s)ds, \quad
 0\leq j\leq q-1,\\
\gamma_jv^{(2j)}(1)-\delta_jv^{(2j+1)}(1)=\int_0^1\psi_j(s)v^{(2j)}(s)ds, \quad 
0\leq j\leq q-1,
\end{gather*}
where $0<t<1,a_i\in((0,1),[0,+\infty))$, $a_i(t)$ are allowed to be singular
at $t=0$ or $t=1$, $i=1,2$.

Anand et al. \cite{Putcha} addressed the question of the existence of at least
 three symmetric positive solutions for the system of dynamical equations 
on symmetric times scales
\begin{gather*}
  (-1)^ny_1^{(\Delta\nabla)^n}=f_1(t,y_1,y_2), \quad t\in[a,b]_{\mathbb{T}}\,,\\
  (-1)^my_2^{(\Delta\nabla)^m}=f_2(t,y_1,y_2), \quad  t\in[a,b]_{\mathbb{T}}\,,
\end{gather*}
subject to the two-point boundary conditions
\begin{gather*}
  y_1^{(\Delta\nabla)^i}(a)=0=y_1^{(\Delta\nabla)^i}(b),\quad
  i=0,1,2,\dots,n-1,\\
  y_2^{(\Delta\nabla)^j}(a)=0=y_2^{(\Delta\nabla)^j}(b),\quad
  j=0,1,2,\dots,m-1,
\end{gather*}
where $f_i:[a,b]_{\mathbb{T}}\times\mathbb{R}^{2}\to [0,\infty)$ are
continuous and $f_i(t,y_1,y_2)=f_i(a+b-t,y_1,y_2)$ for $i=1,2$,
$a\in \mathbb{T}_{k}$, $b\in\mathbb{T}^{k}$ for a time scale $\mathbb{T}$, 
and $\sigma(a)<\rho(b)$. The main tool in \cite{Putcha} 
is the Avery fixed point theorem, a generalization of the Leggett-Williams 
fixed point theorem.

 Yang et al. \cite{Yang1} studied the existence, multiplicity, and uniqueness 
of positive solutions for the boundary value problem
\begin{equation}\label{introyang1}
\begin{gathered}
(-1)^nu^{(2n)}=f(t, u, u',\dots, (-1)^{n-1}u^{(2n-1)}),\\
u^{(2i)}(0)=u^{(2i+1)}(1)=0(j=0,1,\dots, n-1),
\end {gathered}
\end{equation}
where $n\geq 2$, and $f\in C([0,1]\times \mathbb{R}_+^{n+1},\mathbb{R}_+)$. 
The main results obtained in \cite{Yang1} are presented in terms of
a linear function associated with the nonlinearity $f$ in \eqref{introyang1}.  
They also apply their main results to establish the existence, multiplicity, 
and uniqueness of positive symmetric solutions for a Lidstone problem involving 
an open question posed by  Eloe in 2000.


However, the existence problem of positive solutions for systems,
like \eqref{higher-order},   has not been extensively studied yet.
Our main difficulty here arises from the presence of all derivatives of
all odd orders in the nonlinearities $f$ and $g$ in
\eqref{higher-order}. To overcome this difficulty,
as in \cite{Yang2}, we first use the method of order reduction to
transform \eqref{higher-order} into an equivalent system of
integro-differential equations, then prove the existence and
multiplicity of positive solutions for the resultant  equivalent
system, thereby establishing  our main results
for \eqref{higher-order}. Our main features are threefold. Firstly,
the nonlinear functions $f$ and $g$ contain all derivatives of odd
orders. Secondly, nonnegative matrices are used to obtain the priori
estimates of positive solutions. Finally,  the orders $2m$ and $2n$
in \eqref{higher-order} may be different. Such problems can be found
in applied sciences; see \cite{Lazer}.

This paper is organized as follows. Section 2 contains  some
preliminary results, including some basic facts recalled from
\cite{Yang1}. Our main results, namely
Theorems \ref{thm3.1}--\ref{thm3.3}, are stated and proved in 
Section 3. Finally, three examples that illustrate our main
results are presented in Section 4.


\section{Preliminaries}

 Let
\[
E:=C^1([0,1],\mathbb{R}), \|u\|:=\max\{\|u\|_{0},\|u'\|_{0}\},
\]
where $\|u\|_{0}=\max\{|u(t)|:t\in[0,1]\}$. 
Furthermore, put
\[
P:=\{u\in E: u(t)\geq 0,u'(t)\geq 0, \forall t\in
[0,1]\}.
\] 
Clearly, $(E, \|\cdot\|)$ is a real Banach space and  $P$
is a cone in $E$. Let
\[
k(t,s):=\min\{t,s\},(Tu)(t):=\int_0^1k(t,s)u(s)ds.
\] 
Now let $ u:=(-1)^{m-1}w^{(2m-2)},v:=(-1)^{n-1}z^{(2n-2)}$. Then
\eqref{higher-order} is equivalent to the system of
integro-differential equations
\begin{equation}\label{integro-or1}
\begin{gathered}
-u''=f(t, T^{m-1}u,(T^{m-1}u)', \dots,
(Tu)',u', T^{n-1}v,(T^{n-1}v)', \dots,
(Tv)',v'),\\
-v''=g(t, T^{m-1}u,(T^{m-1}u)', \dots,
(Tu)',u', T^{n-1}v,(T^{n-1}v)', \dots,
(Tv)',v'),\\
u(0)=u'(1)=0,\\
v(0)=v'(1)=0.
\end{gathered}
\end{equation}
Furthermore, the above problem is equivalent to
\begin{equation}\label{integro-or2}
\begin{aligned}
u(t)&=\int_0^1k(t,s)f(s, (T^{m-1}u)(s),(T^{m-1}u)'(s),
\dots,
(Tu)'(s),u'(s),\\
 &\quad (T^{n-1}v)(s),(T^{n-1}v)'(s), \dots,
(Tv)'(s),v'(s))ds,\\
v(t)&= \int_0^1k(t,s)g(s, (T^{m-1}u)(s),(T^{m-1}u)'(s),
\dots,
(Tu)'(s),u'(s),\\
 &\quad (T^{n-1}v)(s),(T^{n-1}v)'(s), \dots,
(Tv)'(s),v'(s))ds,
\end{aligned}
\end{equation}
Define the operators $A_i:P\times P\to P$ $(i=1,2)$ and
$A: P\times P \to P\times P$ by 
\begin{align*}
 A_1(u,v)(t)&:=  \int_0^1k(t,s)f(s,(T^{m-1}u)(s),(T^{m-1}u)'(s), \dots,
(Tu)'(s),u'(s),\\
 &\quad (T^{n-1}v)(s),(T^{n-1}v)'(s), \dots,
(Tv)'(s),v'(s))ds,
\\
A_2(u,v)(t)&:=  \int_0^1k(t,s)g(s,(T^{m-1}u)(s),(T^{m-1}u)'(s), \dots,
(Tu)'(s),u'(s),\\
 &\quad (T^{n-1}v)(s),(T^{n-1}v)'(s), \dots, (Tv)'(s),v'(s))ds,
\end{align*}
\[
A(u,v)(t):=(A_{1}(u,v),A_{2}(u,v)).
\]
Now $f\in C([0,1]\times \mathbb{R}_+^{m+n+2},\mathbb{R}_+)$ and 
$g\in C([0,1]\times \mathbb{R}_+^{m+n+2},\mathbb{R}_+)$ imply that $A_i$
and $A$ are completely continuous operators. In our setting,  the
existence of positive solutions for \eqref{higher-order} is
equivalent to that of positive fixed points of $A: P\times P\to P\times P$.
Let
\begin{equation}\label{convention1}
\begin{split}
G_{1}(u,v)(t)&:=  f(t, (T^{m-1}u)(t),(T^{m-1}u)'(t), \dots,
(Tu)'(t),u'(t), (T^{n-1}v)(t),\\
&\quad (T^{n-1}v)'(t), \dots, (Tv)'(t),v'(t))
\end{split}
\end{equation}
\begin{equation}\label{convention2}
\begin{split}
G_{2}(u,v)(t)&:=  g(t, (T^{m-1}u)(t),(T^{m-1}u)'(t), \dots,
(Tu)'(t),u'(t),(T^{n-1}v)(t),\\
&\quad (T^{n-1}v)'(t), \dots, (Tv)'(t),v'(t))
\end{split}
\end{equation}
 Then $ G_i:P\times P \to P$ $(i=1,2)$ is a
continuous, bounded operator, and 
\[
A_i(u,v)(t)=\int_0^1k(t,s)G_i(u,v)(s)ds, i=1,2.
\]

\begin{lemma}[{\cite[Lemma 2.2]{Yang1}}] \label{lem2.1}
 Let $q\in P$. then
 \[
\int_{0}^{1}((T^{n-1}q)(t)+2\sum_{i=0}^{n-1}(T^{n-1-i}q)'(t))te^{t}dt
=\int_{0}^{1}(q(t)+2q'(t))te^{t}dt.
\]
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.3]{Yang1}}]  \label{lem2.2}
 If $ q\in P\bigcap C^{2}[0,1]$,  $q(0)=q'(1)=0$, then
\[
 \int_{0}^{1}(-q''(t))te^{t}dt=\int_{0}^{1}(q(t)+2q'(t))te^{t}\,dt\,.
\]
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.4]{Yang1}}] \label{lem2.3}
If $q\in P$, $q(0)=0$, then 
\[
q(1)\leq \int_{0}^{1}(q(t)+2q'(t))te^{t}\,dt\,.
\]
\end{lemma}

\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
$\omega_{0}\in P\backslash \{0\}$ such that
\[
\omega-T\omega\neq \lambda\omega_{0}, \forall \lambda\geq0,
\omega\in\partial\Omega\cap P, 
\] 
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
\[
  \omega-\lambda T\omega\neq0, \forall \lambda\in[0,1],
  \omega\in\partial\Omega\cap P,
\]
then $i(T,\Omega\cap P,P)=1$.
\end{lemma}

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

A real matrix $B$ is said to be nonnegative if all elements of $B$ are nonnegative.

For the sake of simplicity, we denote by
$x:=(x_{1},\dots,x_{m+1})\in\mathbb{R}^{m+1}_+$,
$y:=(y_{1},\dots,y_{n+1})\in\mathbb{R}^{n+1}_+$. Let
\[
h_{1}(x):=x_{1}+2\sum_{i=2}^{m+1}x_i,\quad
h_{2}(y):=y_{1}+2\sum_{i=2}^{n+1}y_i, \quad x\in\mathbb{R}_+^{m+1},\; 
y\in\mathbb{R}_+^{n+1}.
\]
We now list our hypotheses on $f$ and $g$.
\begin{itemize}

\item[(F1)] $f, g\in C([0,1]\times\mathbb{R}_+^{m+n+2},\mathbb{R}_+)$.

\item[(F2)] There are four nonnegative constants $a_1,a_2,b_1,b_2$, and a
real number $c>0$ such that
\[
 f(t,x,y)\geq a_1h_1(x)+b_1h_2(y)-c,
 g(t,x,y)\geq a_2h_1(x)+b_2h_2(y)-c,
\]
for all $(t,x,y)\in[0,1]\times \mathbb{R}_+^{m+n+2}$ and the matrix
$B_1:=\begin{pmatrix} a_1-1&b_1 \\ a_2&
b_2-1\end{pmatrix} $ is invertible with $B_1^{-1}$ nonnegative.

\item[(F3)] For every $N>0$, there exist two functions $\Phi_N, \Psi_N\in
C(\mathbb{R}_+,\mathbb{R}_+)$ such that
\[
  f(t,x,y)\leq\Phi_N(x_{m+1}+y_{n+1}),
  g(t,x,y)\leq\Psi_N(x_{m+1}+y_{n+1})
\]
for all $(x_1,\dots,x_m)\in\underbrace{[0,N]\times\dots\times[0,N]}_m$,
$(y_1,\dots,y_n)\in\underbrace{[0,N]\times\dots\times [0,N]}_n$ and 
$x_{m+1},y_{n+1}\geq0$, and
\[
   \int_0^{\infty}\frac{\tau
   d\tau}{\Phi_N(\tau)+\Psi_N(\tau)+\delta}=\infty
\]
for all $\delta>0$.

\item[(F4)] There are four nonnegative constants $c_1, c_2, d_1, d_2$ and
a positive constant $r$ such that
\[
  f(t,x,y)\leq c_1 h_1(x)+d_1 h_2(y), \quad
  g(t,x,y)\leq c_2 h_1(x)+d_2 h_2(y)
\]
for all $(t,x,y)\in[0,1]\times([0,r])^{m+n+2}$ and 
$B_2:=\begin{pmatrix} 1-c_1&-d_1 \\ -c_2&
1-d_2\end{pmatrix} $ is invertible with $B_2^{-1}$ nonnegative.



\item[(F5)] There are four nonnegative constants $l_1,l_2,m_1,m_2$ and
a postive constant $c$ such that
\[
 f(t,x,y)\leq l_1 h_1(x)+m_1 h_2(y)+c,
 g(t,x,y)\leq l_2 h_1(x)+m_2 h_2(y)+c,
\]
for all $(t,x,y)\in[0,1]\times\mathbb {R}_+^{m+n+2}$ and 
$B_3:=\begin{pmatrix} 1-l_1&-m_1 \\ -l_2&
1-m_2\end{pmatrix} $ is invertible with $B_3^{-1}$ nonnegative.

\item[(F6)] There are four nonnegative constants $p_1,p_2,q_1,q_2$ and a
positive constant $r$ such that
\[
  f(t,x,y)\geq p_1h_1(x)+q_1h_2(y),
  g(t,x,y)\geq p_2h_1(x)+q_2h_2(y),
\]
for all $(t,x,y)\in[0,1]\times([0,r])^{m+n+2}$ and 
$B_4:=\begin{pmatrix} p_1-1&q_1 \\ p_2&
q_2-1\end{pmatrix} $ is invertible with $B_4^{-1}$ nonnegative.

\item[(F7)] $f(t,x,y)$ and $g(t,x,y)$ are increasing in $x$ and $y$,and there is a constant $\Lambda>0$ such that
\[
  \int_0^1f(s,\underbrace{\Lambda,\dots,\Lambda}_{m+n+2})ds<\Lambda,
  \int_0^1g(s,\underbrace{\Lambda,\dots,\Lambda}_{m+n+2})ds<\Lambda.
\]
\end{itemize}

\begin{remark}[{\cite[Remark 2]{Yang2}}] \label{rmk1} \rm
 Let $l_{ij}(i,j=1,2)$ be four nonnegative
 constants. Then it is easy to see that the matrix 
$B:=\begin{pmatrix}l_{11}-1&l_{12} \\ l_{21}&
l_{22}-1\end{pmatrix}$ is invertible with $B^{-1}$ nonnegative
if and only if one of the following two conditions is satisfied:
\begin{itemize}
\item[(1)] $l_{11}> 1$, $l_{22}>1$, $l_{12}=l_{21}=0$.

\item[(2)] $l_{11}\leq1$, $l_{22}\leq1$, 
$\det B=(1-l_{11})(1-l_{22})-l_{12}l_{21}<0$.
\end{itemize}
\end{remark}

\begin{remark}[{\cite[Remark 3]{Yang2}}]  \label{rmk2} \rm
Let $l_{ij}(i,j=1,2)$ be four nonnegative
 constants. Then it is easy to see that the matrix 
$D:=\begin{pmatrix} 1-l_{11}&-l_{12} \\ -l_{21}&
1-l_{22}\end{pmatrix}$ is invertible with $D^{-1}$ nonnegative
if and only if $l_{11}<1,l_{22}<1,\det
D=(1-l_{11})(1-l_{22})-l_{12}l_{21}>0$.
\end{remark}

\begin{remark} \label{rmk3}\rm
 $f(t,x,y)$ is said to be increasing in $x$ and $y$ if $f(t,x,y)\leq f(t,x',y')$ 
holds for every pair $(x,y),(x',y')\in \mathbb {R}^{m+n+2}_+$ with $(x,y)\leq(x',y')$, 
where the partial ordering $\leq$ in $\mathbb{R}_+^{m+n+2}$ is understood componentwise.
\end{remark}

We have the following comments about the functions $f$ and $g$.
\begin{itemize}
\item[(1)] Condition (F3) is of Berstein-Nagumo type;

\item[(2)] $f$ and $g$ grow superlinearly both at $+\infty$ and 
at $0$ if (F2) and (F4) hold;

\item[(3)] $f$ and $g$  grow sublinearly both at $+\infty$ 
and at $0$ if  (F5) and (F6) hold.
\end{itemize}

We adopt the convention in the sequel that $n_1,n_2,\dots$ stand
for different positive constants. $G_1$ and $G_2$ are defined by
\eqref{convention1} and \eqref{convention2}.

\begin{theorem} \label{thm3.1}
If {\rm (F1)--(F4)} hold, then
\eqref{higher-order} has at least one positive solution.
\end{theorem}

\begin{proof}
 It suffices to prove that
\eqref{integro-or2} has at least one positive solution. We
claim that the set 
\[
\mathcal {M}_{1}:=\{(u,v)\in P\times P:
(u,v)=A(u,v)+\lambda(\varphi,\varphi), \lambda\geq0\}
\] 
is bounded, where $\varphi(t):=te^{-t}$.  Indeed, if
$(u_{0},v_{0})\in\mathcal{M}_{1}$, then there exist a constant
$\lambda_{0}\geq0$ such that
$(u_{0},v_{0})=A(u_{0},v_{0})+\lambda_{0}(\varphi,\varphi)$, which
can be written in the form
\[
  -u_0''(t)=G_1(u_0,v_0)(t)+\lambda_0(2-t)e^{-t},
  -v_0''(t)=G_2(u_0,v_0)(t)+\lambda_0(2-t)e^{-t}.
\]
By (F2), we have
\begin{align*}
-u_0''(t)&\geq  a_1((T^{m-1}u_0)(t)+2\sum_{i=0}^{m-1}(T^{m-i-1}u_0)'(t))+b_1((T^{n-1}v_0)(t)\\
 &\quad +2\sum_{i=0}^{n-1}(T^{n-i-1}v_0)'(t))-c,
\end{align*}
\begin{align*}
 -v_0''(t)&\geq
  a_2((T^{m-1}u_0)(t)+2\sum_{i=0}^{m-1}(T^{m-i-1}u_0)'(t))+b_2((T^{n-1}v_0)(t)\\
 &\quad +2\sum_{i=0}^{n-1}(T^{n-i-1}v_0)'(t))-c.
\end{align*}
Multiply by $\psi(t):=te^{t}$ on both sides of the last two
inequalities and integrate over $[0,1]$, and use Lemmas \ref{lem2.1} and 
\ref{lem2.2}
to obtain
\[
  \int_0^1(u_0(t)+2u_0'(t))te^tdt\geq
  a_1\int_0^1(u_0(t)+2u_0'(t))te^tdt+b_1\int_0^1(v_0(t)+2v_0'(t))te^tdt-c
\]
and
\[
   \int_0^1(v_0(t)+2v_0'(t))te^tdt\geq
  a_2\int_0^1(u_0(t)+2u_0'(t))te^tdt+b_2\int_0^1(v_0(t)+2v_0'(t))te^tdt-c.
\]
The above two inequalities can be written as
\begin{align*}
  \begin{pmatrix} a_1-1&b_1 \\a_2&b_2-1 \end{pmatrix} 
  \begin{pmatrix}\int_0^1(u_0(t)+2u_0'(t))te^tdt \\
 \int_0^1(v_0(t)+2v_0'(t))te^tdt
  \end{pmatrix} 
&=B_1 \begin{pmatrix}\int_0^1(u_0(t)+2u_0'(t))te^tdt \\\int_0^1(v_0(t)+2v_0'(t))te^tdt
  \end{pmatrix} \\
&\leq\begin{pmatrix}c \\c
  \end{pmatrix} .
\end{align*}
Now (F2) implies
\[
  \begin{pmatrix}\int_0^1(u_0(t)+2u_0'(t))te^tdt \\\int_0^1(v_0(t)+2v_0'(t))te^tdt
  \end{pmatrix} \leq B_1^{-1}\begin{pmatrix}c \\c
  \end{pmatrix} :=\begin{pmatrix}N_1 \\ N_2
  \end{pmatrix} .
\]
Let $N:=\max\{N_1, N_2\}>0$.  Then we have
\[
  \int_0^1(u_0(t)+2u_0'(t))te^tdt\leq N, \quad
  \int_0^1(v_0(t)+2v_0'(t))te^tdt\leq N,  \quad \forall
  (u_0,v_0)\in\mathcal {M}_{1}.
\]
Now Lemma \ref{lem2.3} implies
\begin{equation}\label{ethm1}
 \|u_0\|_{0}=u_0(1)\leq N, \|v_0\|_{0}=v_0(1)\leq N, \quad \forall
 (u_0,v_0)\in\mathcal {M}_{1}.
\end{equation}
Furthermore, this estimate leads to
\[
  \|T^{m-1}u_0\|_0=(T^{m-1}u_0)(1)\leq N,
  \|T^{n-1}v_0\|_0=(T^{n-1}v_0)(1)\leq N,
\]
for all $(u_0,v_0)\in\mathcal {M}_{1}$ and
\begin{gather*}
  \|(T^{m-i-1}u_0)'\|_0=(T^{m-i-1}u_0)'(0)=\int_0^1(T^{m-i-2}u_0)(t)dt\leq
  N,
\\
  \|(T^{n-j-1}v_0)'\|_0=(T^{n-j-1}v_0)'(0)=\int_0^1(T^{n-j-2}v_0)(t)dt\leq
  N, \forall (u_0,v_0)\in\mathcal {M}_{1},
\end{gather*}
$i=0,\dots,m-2$, $j=0,\dots,n-2$. 
Let
$$
\mathbb{H}:=\{\mu\geq0:\text{ there exists }(u,v)\in P\times P,
\text{ such that }(u,v)=A(u,v)+\mu(\varphi,\varphi)\}.
$$
Now \eqref{ethm1} implies that $\mu_{0}:=\sup\mathbb{H}<+\infty$.
By (F3), there are two functions $\Phi_N, \Psi_N\in C(\mathbb{R}_+,\mathbb{R}_+)$ 
such that
\[
G_1(u,v)(t)\leq\Phi_N(u'(t)+v'(t)), \quad
G_2(u,v)(t)\leq\Psi_N(u'(t)+v'(t)),
 \]
for all $(u,v)\in\mathcal {M}_{1}$. Hence we obtain
\begin{align*} -u''(t)&=G_1(u,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))+\mu(2-t)e^{-t}\\
&\leq\Phi_N(u'(t)+v'(t))+2\mu_0,
\end{align*}
\begin{align*} -v''(t)&=G_2(u,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))+\mu(2-t)e^{-t}\\
&\leq\Psi_N(u'(t)+v'(t))+2\mu_0,
\end{align*} 
so that
\[
  -u''(t)-v''(t)\leq\Phi_N(u'(t)+v'(t))+\Psi_N(u'(t)+v'(t))+4\mu_0
\]
for all $(u,v)\in\mathcal {M}_{1}$, $\mu\in\mathbb{H}$, and
\[
  \int_0^{u'(0)+v'(0)}\frac{\tau d\tau}{\Phi_N(\tau)+\Psi_N(\tau)+4\mu_0}
  \leq\int_0^1u'(t)+v'(t)dt=u(1)+v(1)\leq 2N
\]
for all $(u,v)\in\mathcal {M}_{1}$. By (F3) 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 means that $\mathcal{M}_1$ is bounded. Taking
$R>\sup\{\|(u,v)\|:(u,v)\in\mathcal{M}_1\}$, we have
\[
 (u,v)\neq A(u,v)+\lambda(\varphi,\varphi), \quad \forall
 (u,v)\in\partial\Omega_R\cap(P\times P), \lambda\geq0.
\]
Now Lemma \ref{lem2.4} yields
\begin{equation}\label{ethm2}
  i(A,\Omega_R\cap(P\times P),P\times P)=0.
\end{equation}
Let 
\[
\mathcal{M}_2:=\{(u,v)\in\overline{\Omega}_r\cap(P\times
P):(u,v)=\lambda A(u,v), \lambda\in[0,1]\}.
\]
Now we want to prove that $\mathcal{M}_2=\{0\}$. Indeed, if
$(u,v)\in\mathcal{M}_2$, then there is $\lambda\in[0,1]$ such that
\[
 (u(t),v(t))=(\lambda\int_0^1k(t,s)G_1(u,v)(s)ds,\lambda\int_0^1k(t,s)G_2(u,v)(s)ds)
\]
which can be written in the form
\[
  -u''(t)=\lambda G_1(u,v)(t), \quad 
  -v''(t)=\lambda G_2(u,v)(t).
\]
By (F4), we have
\begin{gather*}
\begin{aligned}
  -u''(t)&\leq
  c_1((T^{m-1}u)(t)+2\sum_{i=0}^{m-1}(T^{m-i-1}u)'(t))+d_1((T^{n-1}v)(t)\\
&\quad +2\sum_{i=0}^{n-1}(T^{n-i-1}v)'(t)), \end{aligned}
\\
\begin{aligned}
 -v''(t)&\leq
  c_2((T^{m-1}u)(t)+2\sum_{i=0}^{m-1}(T^{m-i-1}u)'(t))+d_2((T^{n-1}v)(t)\\
&\quad +2\sum_{i=0}^{n-1}(T^{n-i-1}v)'(t)).
\end{aligned}
\end{gather*}
Multiply by $\psi(t):=te^{t}$ on both sides of the above and integrate
over $[0,1]$, and use Lemmas \ref{lem2.1} and  \ref{lem2.2} to obtain
\begin{align*}
  \begin{pmatrix} 1-c_1&-d_1 \\-c_2&1-d_2 \end{pmatrix}
  \begin{pmatrix}\int_0^1(u(t)+2u'(t))te^tdt \\ \int_0^1(v(t)+2v'(t))te^tdt
  \end{pmatrix}
& =B_2 \begin{pmatrix}\int_0^1(u(t)+2u'(t))te^tdt \\ \int_0^1(v(t)+2v'(t))te^tdt
  \end{pmatrix} \\
& \leq\begin{pmatrix}0 \\0
  \end{pmatrix} .
\end{align*}
(F4) again implies
\[
  \begin{pmatrix}\int_0^1(u(t)+2u'(t))te^tdt \\ \int_0^1(v(t)+2v'(t))te^tdt
  \end{pmatrix} \leq B_2^{-1}\begin{pmatrix}0 \\0
  \end{pmatrix} =\begin{pmatrix}0 \\0
  \end{pmatrix} .
\]
Consequently,
$$
\int_0^1(u(t)+2u'(t))te^tdt=\int_0^1(v(t)+2v'(t))te^tdt=0
$$
and whence $u\equiv0$, $v\equiv0$, as required. Thus 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
\[
  i(A,\Omega_r\cap(P\times P),P\times   P)=1.
\]
This together with \eqref{ethm2} implies
\[
  i(A,(\Omega_R\backslash\overline{\Omega}_r)\cap(P\times P),P\times P)=0-1=-1.
\]
Therefore, $A$ has at least one fixed point on
$(\Omega_R\backslash\overline{\Omega}_r)\cap(P\times P)$ and
\eqref{integro-or2} has at least one positive solution
$(u,v)$, and thus \eqref{higher-order} has at least one positive
solution $(w,z)=(T^{m-1}u,T^{n-1}v)$. This completes the proof.
\end{proof}

\begin{theorem} \label{thm3.2} 
If {\rm (F1), (F5), (F6)} hold, then \eqref{higher-order} has at least one
positive solution.
\end{theorem}

\begin{proof}
 Let
\[
  \mathcal{M}_{3}:=\{(u,v)\in P\times P:(u,v)=\lambda A(u,v),
  \lambda\in[0,1]\}.
\]
We now assert that $\mathcal{M}_3$ is bounded. Indeed,  if
$(u,v)\in\mathcal{M}_{3}$, then there is $\lambda\in[0,1]$ such that
\[
  u(t)=\lambda\int_0^1k(t,s)G_1(u,v)(s)ds,
  v(t)=\lambda\int_0^1k(t,s)G_2(u,v)(s)ds,
\]
which can be written in the form
\[
  -u''(t)=\lambda G_1(u,v)(t), \quad   -v''(t)=\lambda G_2(u,v)(t).
\]
By (F5), we have
\begin{equation}\label{ethm3}
\begin{aligned}
-u''(t)&\leq
l_1((T^{m-1}u)(t)+2\sum_{i=0}^{m-1}(T^{m-i-1}u)'(t))+m_1((T^{n-1}v)(t)\\
&\quad +2\sum_{i=0}^{n-1}(T^{n-i-1}v)'(t))+c
\end{aligned}\end{equation}
and
\begin{equation}\label{ethm4}
\begin{aligned} 
-v''(t)&\leq l_2((T^{m-1}u)(t)+2\sum_{i=0}^{m-1}(T^{m-i-1}u)'(t))+m_2((T^{n-1}v)(t)\\
&\quad +2\sum_{i=0}^{n-1}(T^{n-i-1}v)'(t))+c.
\end{aligned} 
\end{equation}
Multiply by $\psi(t):=te^{t}$ on both sides of the above two
inequalities and integrate over $[0,1]$, and then use 
Lemmas \ref{lem2.1} and \ref{lem2.2}
to obtain
\[
  \int_0^1(u(t)+2u'(t))te^tdt\leq
  l_1\int_0^1(u(t)+2u'(t))te^tdt+m_1\int_0^1(v(t)+2v'(t))te^tdt+c
\]
and
\[
  \int_0^1(v(t)+2v'(t))te^tdt\leq
  l_2\int_0^1(u(t)+2u'(t))te^tdt+m_2\int_0^1(v(t)+2v'(t))te^tdt+c,
\]
which can be written in the form
\begin{align*}
  \begin{pmatrix} 1-l_1&-m_1 \\-l_2&1-m_2 \end{pmatrix}
  \begin{pmatrix}\int_0^1(u(t)+2u'(t))te^tdt \\\int_0^1(v(t)+2v'(t))te^tdt
  \end{pmatrix} 
&=B_3 \begin{pmatrix}\int_0^1(u(t)+2u'(t))te^tdt \\\int_0^1(v(t)+2v'(t))te^tdt
  \end{pmatrix} \\
&\leq\begin{pmatrix}c \\c
  \end{pmatrix} .
\end{align*}
(F5) again implies
\[
  \begin{pmatrix}\int_0^1(u(t)+2u'(t))te^tdt \\\int_0^1(v(t)+2v'(t))te^tdt
  \end{pmatrix} \leq B_3^{-1}\begin{pmatrix}c \\c
  \end{pmatrix} :=\begin{pmatrix}n_3 \\n_4
  \end{pmatrix} .
\]
Let $N=\max\{n_3,n_4\}>0$. Then we have
\[
  \int_0^1(u(t)+2u'(t))te^tdt\leq N,
  \int_0^1(v(t)+2v'(t))te^tdt\leq N, \forall
  (u,v)\in\mathcal{M}_3.
\]
By Lemma \ref{lem2.3}, we obtain
\begin{gather*}
  \|u\|_0=u(1)\leq\int_0^1(u(t)+2u'(t))te^tdt\leq N,\\
  \|v\|_0=v(1)\leq\int_0^1(v(t)+2u'(t))te^tdt\leq N,
\end{gather*}
for all $(u,v)\in\mathcal{M}_3$. Furthermore, those estimates lead
to
\[
  \|T^{m-1}u\|_0=(T^{m-1})(1)\leq N, \quad
  \|T^{n-1}u\|_0=(T^{n-1})(1)\leq N,
\]
for all $(u,v)\in\mathcal{M}_3$ and
\begin{gather*}
  \|(T^{m-i-1}u)'\|_0=(T^{m-i-1}u)'(0)=\int_0^1(T^{m-i-2}u)(t)dt\leq
  N,
\\
  \|(T^{n-j-1}v)'\|_0=(T^{n-j-1}v)'(0)=\int_0^1(T^{n-j-2}v)(t)dt\leq
  N
\end{gather*}
for all $(u,v)\in\mathcal{M}_3$ and $i=0,\dots,m-2$,
$j=0,\dots,n-2$. By \eqref{ethm3} and \eqref{ethm4}, we have
\begin{gather*}
  -u''(t)\leq
  (l_1(2m-1)+m_1(2n-1))N+2l_1u'(t)+2m_1v'(t)+c, \\
  -v''(t)\leq
  (l_2(2m-1)+m_2(2n-1))N+2l_2u'(t)+2m_2v'(t)+c,
\end{gather*}
for all $(u,v)\in\mathcal{M}_3$. So we have
\begin{align*} 
-(u''(t)+v''(t))&\leq
(l_1(2m-1)+m_1(2n-1)+l_2(2m-1)+m_2(2n-1))N\\
&\quad +2(l_1+l_2)u'(t)+2(m_1+m_2)v'(t)+2c.
\end{align*}
Noticing $u'(1)=v'(1)=0$ and letting
\[
N_2:=(l_1(2m-1)+m_1(2n-1)+l_2(2m-1)+m_2(2n-1))N+2c
\]
and $L:=2(l_1+l_2+m_1+m_2)+1$, we obtain
\[
  u'(t)+v'(t)\leq\frac{N_2}{L}(e^{L-Lt}-1),
\]
so that
\[
  \|u'+v'\|_0=u'(0)+v'(0)\leq
  \frac{N_2}{L}(e^L-1).
\]
This proves the boundedness of $\mathcal{M}_3$. Taking
$R>\sup\{\|(u,v)\|:(u,v)\in\mathcal{M}_3\}$, 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{ethm5}
  i(A,\Omega_R\cap(P\times P),P\times P)=1.
\end{equation}
Let
\[
  \mathcal{M}_4:=\{(u,v)\in \overline{\Omega}_r\cap (P\times P):
  (u,v)=A(u,v)+\lambda(\varphi,\varphi),\lambda\geq 0\}
\]
where $\varphi(t):=te^{-t}$. We want to prove that
$\mathcal{M}_4\subset\{0\}$. Indeed, if $(u,v)\in\mathcal{M}_4$,
then there is $\lambda\geq0$ such that
\[
  u(t)=\int_0^1k(t,s)G_1(u,v)(s)ds+\lambda\varphi(t), \quad 
  v(t)=\int_0^1k(t,s)G_2(u,v)(s)ds+\lambda\varphi(t),
\]
which can be written in the form
\[
  -u''(t)=G_1(u,v)(t)+\lambda(2-t)e^{-t}, \quad
  -v''(t)=G_2(u,v)(t)+\lambda(2-t)e^{-t}.
\]
By (F6), we have
\begin{align*}
&-u''(t)\\
&\geq  p_1\big((T^{m-1}u)(t)+2\sum_{i=0}^{m-1}(T^{m-i-1}u)'(t)\big)
 +q_1\big((T^{n-1}v)(t)+2\sum_{i=0}^{n-1}(T^{n-i-1}v)'(t)\big),
 \\
&-v''(t)\\
&\geq   p_2\big((T^{m-1}u)(t)+2\sum_{i=0}^{m-1}(T^{m-i-1}u)'(t)\big)
+q_2\big((T^{n-1}v)(t)+2\sum_{i=0}^{n-1}(T^{n-i-1}v)'(t)\big).
\end{align*}
Multiply by $\psi(t):=te^{t}$ on both sides of the above two
inequalities and integrate over $[0,1]$ and use Lemmas \ref{lem2.1} and \ref{lem2.2}
to obtain
\[
  \int_0^1(u(t)+2u'(t))te^tdt\geq
  p_1\int_0^1(u(t)+2u'(t))te^tdt+q_1\int_0^1(v(t)+2v'(t))te^tdt
\]
and
\[
  \int_0^1(v(t)+2v'(t))te^tdt\geq
  p_2\int_0^1(u(t)+2u'(t))te^tdt+q_2\int_0^1(v(t)+2v'(t))te^tdt,
\]
which can be written in the form
\begin{align*}
  \begin{pmatrix} p_1-1&q_1 \\p_2&q_2-1 \end{pmatrix}
  \begin{pmatrix}\int_0^1(u(t)+2u'(t))te^tdt \\ \int_0^1(v(t)+2v'(t))te^tdt
  \end{pmatrix} 
&=B_4 \begin{pmatrix}\int_0^1(u(t)+2u'(t))te^tdt \\\int_0^1(v(t)+2v'(t))te^tdt
  \end{pmatrix} \\
&\leq\begin{pmatrix}0 \\0
  \end{pmatrix} .
\end{align*}
Now (F6) implies
\[
  \begin{pmatrix}\int_0^1(u(t)+2u'(t))te^tdt \\\int_0^1(v(t)+2v'(t))te^tdt
  \end{pmatrix} \leq B_4^{-1}\begin{pmatrix}0 \\0
  \end{pmatrix} =\begin{pmatrix}0 \\0
  \end{pmatrix} .
\]
Therefore,
$\int_0^1(u(t)+2u'(t))te^tdt=\int_0^1(v(t)+2v'(t))te^tdt=0$
and $u\equiv0$, $v\equiv0$. This implies
$\mathcal{M}_4\subset\{0\}$, as required. As a result of this, we
obtain
\[
  (u,v)\neq A(u,v)+\lambda(\varphi,\varphi), \quad \forall
  (u,v)\in\partial\Omega_r\cap(P\times P),\; \lambda\geq0.
\]
Now Lemma \ref{lem2.4} yields
\begin{equation}\label{ethm6}
i(A,\Omega_r\cap(P\times P),P\times P)=0.
\end{equation}
Combining \eqref{ethm5} and \eqref{ethm6} gives
\[
i(A,(\Omega_R\backslash\overline{\Omega}_r)\cap(P\times P),P\times
P)=1.
\]
Hence $A$ has at least one fixed point on
$(\Omega_R\backslash\overline{\Omega}_r)\cap(P\times P)$. Thus
\eqref{higher-order} has at least one positive solution
$(w,z)=(T^{m-1}u,T^{n-1}v)$. This complete the proof.
\end{proof}

\begin{theorem} \label{thm3.3} 
If {\rm (F1)--(F3), (F6), (F7)} hold, then \eqref{higher-order} has at least two
positive solutions.
\end{theorem}

\begin{proof} 
By (F7), the following inequalities
\[
 f(t,x,y)\leq f(t,\underbrace{\Lambda,\dots,\Lambda}_{m+n+2})<\Lambda,\quad 
 g(t,x,y)\leq g(t,\underbrace{\Lambda,\dots,\Lambda}_{m+n+2})<\Lambda,
\]
hold for all $t\in[0,1]$ and all
 $(x,y)\in\underbrace{[0,\Lambda]\times\dots\times [0,\Lambda]}_{m+n+2}$. 
Consequently,
we have for all $(u,v)\in \partial\Omega_{\Lambda}\cap(P\times P)$,
\begin{align*}
\|A_1(u,v)\|_0&=A_1(u,v)(1)=\int_0^1sG_1(u,v)(s)ds\leq\int_0^1G_1(u,v)(s)ds\\
&\leq\int_0^1f(s,\Lambda,\dots,\Lambda)ds<\Lambda=\|(u,v)\|,
\end{align*}
\begin{align*}
\|A_2(u,v)\|_0&=A_2(u,v)(1)=\int_0^1sG_2(u,v)(s)ds\leq\int_0^1G_2(u,v)(s)ds\\
& \leq\int_0^1g(s,\Lambda,\dots,\Lambda)ds<\Lambda=\|(u,v)\|,
\end{align*}
\begin{align*}
\|(A_1(u,v))'\|_0&=(A_1(u,v))'(0)=\int_0^1G_1(u,v)(s)ds\\
&\leq\int_0^1f(s,\Lambda,\dots,\Lambda)ds<\Lambda=\|(u,v)\|,
\end{align*}
\begin{align*}
\|(A_2(u,v))'\|_0&=(A_2(u,v))'(0)=\int_0^1G_2(u,v)(s)ds\\
&\leq\int_0^1g(s,\Lambda,\dots,\Lambda)ds<\Lambda=\|(u,v)\|.
\end{align*}
The preceding inequalities imply $\|A(u,v)\|=\|(A_1(u,v),A_2(u,v))\|<\Lambda=\|(u,v)\|$,
and thus
\[
  (u,v)\neq\lambda A(u,v), \quad
\forall (u,v)\in\partial\Omega_{\Lambda}\cap(P\times P),\; 0\leq\lambda\leq1.
\]
Now Lemma \ref{lem2.5} yields
\begin{equation}\label{ethm7}
  i(A,\Omega_{\Lambda}\cap(P\times P),P\times P)=1.
\end{equation}
By (F2), (F3) and (F6), we know that \eqref{ethm2} and
\eqref{ethm6} hold. Note we can choose $R>\Lambda>r$ in
\eqref{ethm2} and \eqref{ethm6} (see the proofs of
Theorems \ref{thm3.1} and \ref{thm3.2}). Combining \eqref{ethm2}, \eqref{ethm6} and
\eqref{ethm7}, we obtain
\[
i(A,(\Omega_{R}\backslash\overline{\Omega}_{\Lambda})\cap(P\times P),P\times P)=0-1=-1,
\]
and
\[
i(A,(\Omega_{\Lambda}\backslash\overline{\Omega}_{r})\cap(P\times P),P\times P)=1-0=1.
\]
Therefore, $A$ has at least two fixed points, with one on 
$(\Omega_{R}\backslash\overline{\Omega}_{\Lambda})\cap(P\times P)$
and the other on $(\Omega_{\Lambda}\backslash\overline{\Omega}_{r})\cap(P\times P)$.
 Hence \eqref{higher-order} has at least two positive solutions.
\end{proof}

\section{Examples}

In this section we present three examples that  illustrate our main results.

\begin{example} \label{examp4.1} \rm
 Suppose   $(\xi_{ij})_{2\times (m+1)}$ and $(\eta_{ij})_{2\times (n+1)}$ be 
two positive matrices and $1<\alpha_i\leq 2(i=1,2.)$. Let
\begin{gather*}
f(t,x,y):=\Big(\sum_{j=1}^{m+1} \xi_{1j}x_j+\sum_{j=1}^{n+1} \eta_{1j}y_j\Big)^{\alpha_1}
\quad t\in [0,1],\;x\in\mathbb{R}_+^{m+1},\;y\in\mathbb{R}_+^{n+1},
\\
g(t,x,y):=\Big(\sum_{j=1}^{m+1} \xi_{2j}x_j+\sum_{j=1}^{n+1} 
\eta_{2j}y_j\Big)^{\alpha_2} 
\quad t\in [0,1],\;x\in\mathbb{R}_+^{m+1},\;y\in\mathbb{R}_+^{n+1}.
\end{gather*}
Now  (F1)--(F4) hold. By Theorem \ref{thm3.1}, \eqref{higher-order} has at 
least one
positive solution.
\end{example}

\begin{example} \label{examp4.2} \rm
Suppose   $(\xi'_{ij})_{2\times (m+1)}$ and $(\eta'_{ij})_{2\times (n+1)}$ 
be two positive matrices and $0<\alpha_i<1(i=3,4.)$. Let
\begin{gather*}
f(t,x,y):=\Big(\sum_{j=1}^{m+1} \xi_{1j}'x_j
+\sum_{j=1}^{n+1} \eta_{1j}'y_j\Big)^{\alpha_3} 
\quad t\in [0,1],\;x\in\mathbb{R}_+^{m+1},\; y\in\mathbb{R}_+^{n+1},
\\
g(t,x,y):=\Big(\sum_{j=1}^{m+1} \xi_{2j}'x_j+\sum_{j=1}^{n+1} \eta_{2j}'y_j\Big)^{\alpha_4} 
\quad t\in [0,1],\;x\in\mathbb{R}_+^{m+1},\; y\in\mathbb{R}_+^{n+1}.
\end{gather*}
Now  (F1), (F5) and (F6) are satisfied. By Theorem \ref{thm3.2}, \eqref{higher-order} has
 at least one positive solution.
\end{example}

\begin{example} \label{examp4.3} \rm
Suppose $(\xi_{ij})_{2\times (m+1)}$, $(\xi'_{ij})_{2\times (m+1)}$, 
$(\eta_{ij})_{2\times (n+1)}$ and $(\eta'_{ij})_{2\times (n+1)}$ be four
positive matrices, $1<\beta_i\leq2(i=1,2)$, $0<\gamma_i<1$ $(i=1,2)$. Let
\begin{gather*}
f(t,x,y):=\Big(\sum_{j=1}^{m+1} \xi_{1j}x_j+\sum_{j=1}^{n+1} \eta_{1j}y_j\Big)^{\beta_1}
+\Big(\sum_{j=1}^{m+1} \xi_{1j}'x_j+\sum_{j=1}^{n+1} \eta_{1j}'y_j\Big)^{\gamma_1} \\
t\in [0,1],x\in\mathbb{R}_+^{m+1},y\in\mathbb{R}_+^{n+1},
\\
g(t,x,y):=\Big(\sum_{j=1}^{m+1} \xi_{2j}x_j+\sum_{j=1}^{n+1} \eta_{2j}y_j\Big)^{\beta_2}
+\Big(\sum_{j=1}^{m+1} \xi_{2j}'x_j+\sum_{j=1}^{n+1} \eta_{2j}'y_j\Big)^{\gamma_2} \\
t\in [0,1],x\in\mathbb{R}_+^{m+1},y\in\mathbb{R}_+^{n+1}.
\end{gather*}
Now  (F1)-(F3), (F6) and (F7) are satisfied. By Theorem \ref{thm3.3}, \eqref{higher-order} 
has at least two positive solutions.
\end{example}

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\end{document}