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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 11, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/11\hfil Unique solvability]
{Unique solvability for a second order
nonlinear system via  two global inversion theorems}

\author[R. Dalmasso\hfil EJDE-2008/11\hfilneg]
{Robert Dalmasso}

\address{Robert Dalmasso \newline
Laboratoire Jean Kuntzmann\\
Equipe EDP, Tour IRMA, BP 53 \\
38041 Grenoble Cedex 9, France}
\email{robert.dalmasso@imag.fr}

\thanks{Submitted October 16, 2007. Published January 21, 2008.}
\subjclass[2000]{34B15}
\keywords{Nonlinear boundary value problem; existence; uniqueness}

\begin{abstract}
  In this paper we use two global inversion theorems to establish
  the existence and uniqueness for a nonlinear second order
  homogeneous Dirichlet system.
\end{abstract}

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

\section{Introduction}

Let $n \geq 1$ and let $f = (f_1, \dots ,f_n) : [0,
1]\times\mathbb{R}^n \to \mathbb{R}^n$ be a continuous function.
We consider the  system
\begin{equation}
\begin{gathered}
  u''(x) + f(x, u(x)) = 0\,, \quad  0 \leq x \leq 1 ,\\
  u(0) = u(1)  = 0 \,.
\end{gathered}\label{eq:eq1}
\end{equation}
We first introduce some notations:
\begin{displaymath}
\|u\| = \max_{1 \leq j \leq n}(|u_j|), \quad  u = (u_1, \dots ,
u_n) \in \mathbb{R}^n\,,
\end{displaymath}
$M(n)$ is the space of $n\times n$ matrices with real entries and
$\rho(M)$ is the spectral radius of $M \in M(n)$,
\begin{gather*}
  \|M\| = \max_{1 \leq j \leq
n}\sum_{k = 1}^{n}|m_{jk}|, \quad  M = (m_{jk})_{1\leq j,k\leq n}
\in M(n)\,,\\
  \|y\|_p =(\int_0^1{|y(t)|^{p}\, dt})^{1/p},
\quad  y  \in L^p(0,1)\,,\; 1 \leq p < + \infty \,, \\
  \|y\|_{\infty} = \mathop{\rm ess\,sup}_{(0,1)} |y|, \quad
   y  \in L^{\infty}(0,1)\,, \\
  \|y\|_p = \max_{1 \leq j \leq n}(\|y_j\|_p),
\quad  y = (y_1, \dots , y_n) \in L^p((0, 1),\mathbb{R}^n)\,, \;
1 \leq p \leq + \infty, \\
  \mathbb{R}^n_+ = \{x =(x_1,\dots, x_n) \in \mathbb{R}^n; \, x_j \geq 0
  \,, \, j = 1, \dots , n\}\,.
\end{gather*}


Recently the author proved the following theorem.


\begin{theorem}[\cite{da}] \label{thm1}
Assume that the partial derivatives  $\partial f_j/\partial u_k$
exist and are continuous on $[0,1]\times\mathbb{R}^n$ for
$j, k =1, \dots ,n$. Let $\Lambda = (\lambda_{jk})_{1 \leq j, k \leq n}:
\mathbb{R}_+ \to M(n)$ be a continuous map with $\lambda_{jk}$
nondecreasing and bounded for $j, k = 1, \dots , n$. Assume that
\begin{gather}
\bigl|\frac{\partial f_j}{\partial u_k}(x, u)\bigr| \leq
\lambda_{jk}(\|u\|) \quad \forall \, (x, u) \, \in
[0,1]\times\mathbb{R}^n \,, \; 1 \leq j, k \leq n \,,
\label{eq:eq2}
\\
\rho(\Lambda(t)) < \pi^2 \quad \forall \, t \, \geq 0 \,,
\label{eq:eq3}
\\
\int_0^{+ \infty}\det(\pi^2 I - \Lambda(t))\, dt =  + \infty \,.
\label{eq:eq4}
\end{gather}
Then the boundary value problem  \eqref{eq:eq1} has a unique
solution.
\end{theorem}

The purpose of this paper is to improve and complement Theorem \ref{thm1}.
We have the following results.

\begin{theorem} \label{thm2}
  Assume that the partial derivatives  $\partial f_j/\partial u_k$ exist
and are continuous on $[0,1]\times\mathbb{R}^n$ for $j, k = 1, \dots ,n$.
Let $\Lambda = (\lambda_{jk})_{1 \leq j, k \leq n}: \mathbb{R}_+ \to
M(n)$ be a continuous map with $\lambda_{jk}$ nondecreasing
   for $j, k = 1, \dots , n$. Assume that \eqref{eq:eq2} and
\eqref{eq:eq3} hold and that
\begin{equation}
\int_0^{+ \infty}\frac{dt}{\|(\pi^2 I - \Lambda(t))^{- 1}\|} = +
\infty \,. \label{eq:eq5}
\end{equation}
Then the boundary value problem  \eqref{eq:eq1} has a unique
solution.
\end{theorem}

\begin{theorem} \label{thm3}
  Assume that the partial derivatives  $\partial f_j/\partial u_k$
exist and are continuous
on $[0,1]\times\mathbb{R}^n$ for $j, k = 1, \dots ,n$. Let $b \in
\mathbb{R}_+^n$ and let $A = (a_{jk})_{1 \leq j, k \leq n}:
\mathbb{R}_+ \to M(n)$ be a continuous map with $a_{jk}$
nondecreasing for $j, k = 1, \dots , n$. Assume that
\begin{equation}
u_{j}f_{j}(x,u) \leq \sum_{k=1}^{n}a_{jk}(\|u\|)|u_j u_k| +
b_{j}|u_j| \,, \label{eq:eq6}
\end{equation}
for all $(x, u) \, \in [0,1]\times\mathbb{R}^n$,
$1 \leq j \leq n$,
\begin{gather}
\rho(A(t)) < \pi^2 \quad \forall \, t \, \geq 0 \,,
\label{eq:eq7}
\\
\lim_{t\to+\infty}\frac{t}{\|(\pi^2 I - A(t))^{- 1}\|} = + \infty
\,. \label{eq:eq8}
\end{gather}
Let $\Lambda = (\lambda_{jk})_{1 \leq j, k \leq n}: \mathbb{R}_+
\to M(n)$ be a continuous map with $\lambda_{jk}$ nondecreasing
for $j, k = 1, \dots , n$. Assume that \eqref{eq:eq2} and
\eqref{eq:eq3} hold. Then the boundary value problem
\eqref{eq:eq1} has a unique solution.
\end{theorem}

In Section 2 we recall some results from the theory of nonnegative
matrices. We also give two global inversion theorems. We prove
Theorem \ref{thm2} in Section 3 and Theorem \ref{thm3} in section 4. Finally in
Section 5 we conclude with some examples.

\section{Preliminaries}

We begin with some results from the theory of nonnegative
matrices. We refer the reader to \cite{bepl} for proofs.

\begin{definition} \label{def1} \rm
$A \in M(n)$ is called
$\mathbb{R}^n_+$-monotone if
$Ax \in \mathbb{R}^n_+$ implies $x \in \mathbb{R}^n_+$.

$N = (n_{jk})_{1 \leq j, k \leq n}$ is nonnegative if $n_{jk} \geq
0$ for $j, k = 1, \dots, n$.
\end{definition}


\begin{theorem}[{\cite[p. 113]{bepl}}]\label{thm4}
$A \in M(n)$ is $\mathbb{R}^n_+$-monotone if and only if $A$ is nonsingular
and $A^{- 1}$ is nonnegative.
\end{theorem}

\begin{theorem}[{\cite[p. 113]{bepl}}] \label{thm5}
  Let $A = \alpha I - N$ where $\alpha \in \mathbb{R}$ and $N \in M(n)$ is
nonnegative. Then the following are equivalent:
\begin{itemize}
\item[(i)] $A$ is $\mathbb{R}^n_+$-monotone;

\item[(ii)] $\rho(N) < \alpha$.
\end{itemize}
\end{theorem}


\begin{remark} \label{rmk1} \rm
With the notations of Theorem \ref{thm5},
assume that (i) (or (ii)) holds. Then $\det A > 0$.
\end{remark}

The proof of Theorem \ref{thm2} makes use of the following global inversion
theorem of Hadamard-L\'evy type established by M. R\v adulescu and
S. R\v adulescu \cite[Theorem 2]{ra1}.


\begin{theorem} \label{thm6}
  Let $(Y, N_0)$ be a Banach space and let
$L : D(L) \to Y$ be a linear operator with closed graph, where
$D(L)$ is a linear subspace of $Y$. Then $D(L)$ is a Banach space
with respect to the norm defined by
\begin{displaymath}
  N_1(u) = N_0(u) + N_0(Lu) \,, \quad u \in D(L)\,.
\end{displaymath}
Further, let $K : (Y, N_0) \to (Y, N_0)$ be a
$C^1$ map and let $X$ be a linear subspace of $D(L)$ which is
closed in the norm $N_1$. Consider the nonlinear map $ S : (X,
N_1) \to (Y, N_0)$ defined by $S = L - K$, and assume that $S$ is
a local diffeomorphism. If there exists a continuous map $c :
\mathbb{R}_+ \to \mathbb{R}_+^{\star}$ such that
\begin{gather*}
\int_0^{+\infty}{c(t)\, dt} =  + \infty\,, \\
  N_0(S'(u)(h)) \geq c(N_0(u))N_0(h)\quad
\forall \, u\,,\, h \in X \,,
\end{gather*}
then $S$ is a global diffeomorphism.
\end{theorem}

The proof of Theorem \ref{thm3} makes use of  the
Banach-Mazur-Caccioppoli global inversion theorem
(\cite{bm}, \cite{ca} and \cite{ra2}).

\begin{theorem} \label{thm7}
Let $E$ and $F$ be two Banach spaces. Then $S: E \to F$ is a global
homeomorphism if and only if $S$ is a local homeomorphism and a proper
map.
\end{theorem}

\section{Proof of Theorem \ref{thm2}}

We begin with two lemmas.


\begin{lemma} \label{lem1}
Let $w \in C^1([0,1], \mathbb{R})$
be such that $w(0) = w(1) = 0$. Then
\begin{displaymath}
\|w'\|_2 \geq \pi\|w\|_2 \quad \text{and}\quad
\|w'\|_2 \geq 2\|w\|_{\infty}\,.
\end{displaymath}
\end{lemma}

The first inequality is known as the Wirtinger inequality and the
second inequality is known as the Lees inequality.

\begin{lemma} \label{lem2}
  Let
\begin{displaymath} X = \{h \in C^2([0,1],\mathbb{R}^n) ;\, h(0) =
h(1) = 0\} \,,
\end{displaymath} and let $V = (v_{jk})_{1 \leq j,k\leq n}: [0,1]
\to M(n)$ be a continuous map. Assume that there exists $N =
(n_{jk})_{1 \leq j, k \leq n} \in M(n)$ such that $\rho(N) <
\pi^2$ and
\begin{displaymath} |v_{jk}(x)|\leq n_{jk}  \quad \forall \, x \, \in
[0,1] \,, \, \, 1 \leq j, k \leq n \,.
\end{displaymath}
If $T : X \to C([0,1],\mathbb{R}^n)$ is the operator defined by
\begin{displaymath}
T(h)(x) = h''(x) + V(x)h(x) \,, \quad h \in X\,,
\, \, x \in [0,1] \,,
\end{displaymath}
then
\begin{displaymath}
\|T(h)\|_{\infty} \geq
\frac{2}{\pi\|(\pi^2 I - N)^{- 1}\|}\|h\|_{\infty} \quad \forall
\, \, h \in X \,.
\end{displaymath}
\end{lemma}


\begin{proof} Let $h = (h_1, \dots ,h_n) \in X$ and let
$j \in \{1, \dots ,n\}$. Integrating by parts we get
\begin{align*}
\int_0^1{h_j(x)T(h)_j(x)\, dx}
& =  \int_0^1{h_j(x)(h_j''(x) + \sum_{k = 1}^{n}v_{jk}(x)h_k(x))\, dx}
\\
& =   - \int_0^1{h'_j(x)^2\, dx} + \sum_{k=1}^{n}
\int_0^1{v_{jk}(x)h_j(x)h_k(x)\, dx}\,.\\
\end{align*}
Then using the
Cauchy-Schwarz inequality and Lemma \ref{lem1} we can write
\begin{align*}
\|h_j\|_2\|T(h)_j\|_2
& \geq  - \int_0^1{h_j(x)T(h)_j(x)\, dx}  \\
& =   \|h'_j\|_2^2 - \sum_{k=1}^{n} \int_0^1{v_{jk}(x)h_j(x)h_k(x)\, dx}\\
& \geq    \pi\|h_j\|_2\|h'_j\|_2 -
   \sum_{k=1}^{n}n_{jk} \|h_j\|_2 \|h_k\|_2  \\
& \geq  \pi\|h_j\|_2\|h'_j\|_2 - \frac{1}{\pi}
\sum_{k=1}^{n}n_{jk} \|h_j\|_2 \|h'_k\|_2 \,,
  \end{align*}
from which we deduce that
\begin{equation}
\|T(h)_j\|_2  \geq \pi\|h'_j\|_2 -
\frac{1}{\pi}\sum_{k=1}^{n}n_{jk} \|h'_k\|_2  \,, \label{eq:eq9}
\end{equation}
for $j = 1, \dots, n$. Let $a$, $b$ denote the vectors
\begin{displaymath}
a = (\|h'_j\|_2)_{1\leq j \leq n} \quad \text{and}
  \quad b = (\pi \|T(h)_j\|_2)_{1\leq j \leq n}\,.
\end{displaymath}
Inequality \eqref{eq:eq9} can be written
\begin{displaymath}
b - (\pi^2 I - N)a \in \mathbb{R}^n_+\,.
\end{displaymath}
Theorem \ref{thm5} implies that $\pi^2 I - N$ is $\mathbb{R}^n_+$-monotone.
Then using Theorem \ref{thm4} we obtain
\begin{equation}
(\pi^2 I - N)^{- 1}b - a  \in \mathbb{R}^n_+\,, \label{eq:eq10}
\end{equation}
which implies that
\begin{displaymath}
\pi \|(\pi^2 I - N)^{- 1}\|\,\|T(h)\|_2 \geq \|h'_j\|_2 \,,
\end{displaymath}
for $j = 1, \dots , n$. Using Lemma \ref{lem1} and the fact that
$\|T(h)\|_2 \leq \|T(h)\|_{\infty}$ we deduce that
\begin{displaymath}
\|T(h)\|_{\infty} \geq \frac{2}{\pi \|(\pi^2 I - N)^{-
1}\|}\|h\|_{\infty} \,,
\end{displaymath}
and the lemma is proved.
\end{proof}

Now we can complete the proof of Theorem \ref{thm2}. Let
$Y = C([0,1],\mathbb{R}^n)$ be equipped with the sup norm
$\|.\|_{\infty}$ and let $L: D(L) \to Y$ be the linear operator
defined by
\begin{displaymath}
Lu =  u'' \,, \quad u \in D(L) \,,
\end{displaymath}
where $D(L) = C^2([0,1],\mathbb{R}^n)$. Since $L$ has closed
graph, it follows from Theorem \ref{thm6} that $D(L)$ is a Banach space
with respect to the norm $N_1$ defined by
\begin{displaymath}
N_1(u) =  \|u\|_{\infty} + \|Lu\|_{\infty}\,,
\quad u  \in D(L) \,.
\end{displaymath}
  Let $K: (Y, \|.\|_{\infty}) \to (Y, \|.\|_{\infty})$ be
given by
\begin{displaymath}
K(u)(x) = - f(x,u(x)) \,,  \quad u \in Y\,, \, x \in [0,1] \,.
\end{displaymath}
The regularity assumptions on $f$ imply that $K$ is of class
$C^1$. The set $X = \{u \in D(L); u(0) = u(1) = 0\}$ is a closed subspace
of $D(L)$ in the norm $N_1$. Let $S = L - K$. Clearly $S: (X, N_1)
\to (Y, \|.\|_{\infty})$ is of class $C^1$. Let $u \in X$ be fixed
and let $V = (v_{jk})_{1 \leq j, k \leq n}: [0, 1] \to M(n)$ be
such that
\begin{displaymath}
v_{jk}(x) = \frac{\partial f_j}{\partial u_k}(x, u(x))
\,,\quad  x \, \in [0,1] \,, \, \, 1 \leq j, k \leq n \,.
\end{displaymath}
We have
\begin{displaymath}
S'(u)(h)(x) =  h''(x) + V(x)h(x)\,, \quad h
\in X \,, \, x \in [0,1]\,.
\end{displaymath}
Also \eqref{eq:eq2}  implies
\begin{displaymath}
|v_{jk}(x)| \leq \lambda_{jk}(\|u\|_{\infty}) \quad \forall \, x
\, \in [0,1]\,, \, \, 1 \leq j, k \leq n \,.
\end{displaymath}
Then using Lemma \ref{lem2}, we get
\begin{equation}
\|S'(u)(h)\|_{\infty} \geq \frac{2}{\pi\|(\pi^2 I -
\Lambda(\|u\|_{\infty}))^{- 1}\|}\|h\|_{\infty}
   \quad \forall\, h \in X \,.
\label{eq:eq11}
\end{equation}
Let $Q: X \to Y$  be defined by
   \begin{displaymath} Q(h)(x) = - V(x)h(x)\,,\quad h \in X \,, \, x
\in [0,1] \,.
\end{displaymath}
The operator $L: X \to Y$ is one to one and onto.
We have $S'(u) = L - Q = L(I- L^{- 1}Q)$.
By \eqref{eq:eq11} $\ker (S'(u)) = \{0\}$. Then
$\ker(I - L^{- 1}Q) = \{0\}$. Since $L^{- 1}: (Y, \|.\|_{\infty})
\to (X, \|.\|_{\infty})$ is compact, $L^{- 1}Q$ is compact too. By
the Fredholm alternative we obtain that $I - L^{- 1}Q$ is onto.
Therefore $S'(u): (X, N_1) \to (Y, \|.\|_{\infty})$ is an
invertible operator. By the local inversion theorem we have that
$S$ is a local diffeomorphism. Now let
\begin{displaymath}
c(t) = \frac{2}{\pi\|(\pi^2 I - \Lambda(t))^{- 1}\|}\,, \quad t
\geq 0 \,.
\end{displaymath}
This function satisfies the hypotheses of
Theorem \ref{thm6}. Therefore $S$ is a global diffeomorphism and
consequently the equation $Su = 0$ has a unique solution
$u \in X$. This is also the unique solution of the boundary value problem
\eqref{eq:eq1}.

\section{Proof of Theorem \ref{thm3}}

We keep the notations introduced in Section 3. In the same way we
show that $S: (X, N_1) \to (Y, \|\cdot\|_{\infty})$ is a local
diffeomorphism. Now  let $u = (u_1, \dots ,u_n) \in X$ and let
$j\in \{1, \dots ,n\}$. Integrating by parts we get
\begin{align*}
\int_0^1{u_j(x)S(u)_j(x)\, dx}
& =   \int_0^1{u_j(x)(u_j''(x) + f_j(x,u(x))\, dx}\\
& =   - \int_0^1{u'_j(x)^2\, dx} +
\int_0^1{u_j(x)f_j(x,u(x))\, dx}\,.\\
\end{align*}
Then using the Cauchy-Schwarz inequality, \eqref{eq:eq6} and Lemma \ref{lem1}
we can write
\begin{align*}
\|u_j\|_2\|S(u)_j\|_2
& \geq  - \int_0^1{u_j(x)S(u)_j(x)\, dx}  \\
& \geq   \|u'_j\|_2^2 - \sum_{k=1}^{n}
\int_0^1{a_{jk}(\|u(x)\|)|u_j(x)u_k(x)|\, dx}
  - b_j\int_0^1 |u_j(x)|\, dx \\
& \geq    \|u'_j\|_2^2 - \sum_{k=1}^{n}a_{jk}(\|u\|_{\infty})
   \|u_j\|_2 \|u_k\|_2 - b_j\|u_j\|_2 \\
& \geq   \pi\|u_j\|_2\|u'_j\|_2 - \frac{1}{\pi}
  \sum_{k=1}^{n}a_{jk}(\|u\|_{\infty})\|u_j\|_2 \|u'_k\|_2
  - b_j\|u_j\|_2 \,,
   \end{align*}
from which we deduce that
\begin{equation}
\|S(u)_j\|_2  \geq \pi\|u'_j\|_2 -
\frac{1}{\pi}\sum_{k=1}^{n}a_{jk}(\|u\|_{\infty}) \|u'_k\|_2 - b_j
\,, \label{eq:eq12}
\end{equation}
for $j = 1, \dots, n$. Let $r$, $s$ and $b$ denote the vectors
\begin{displaymath}
r = (\|u'_j\|_2)_{1\leq j \leq n}\,, \quad
s = (\pi \|S(u)_j\|_2)_{1\leq j \leq n},  \quad
b = (\pi b_j)_{1\leq j \leq n}.
\end{displaymath}
Inequality \eqref{eq:eq12} can be written as
\begin{displaymath}
s - (\pi^2 I - A(\|u\|_{\infty}))r + b \in \mathbb{R}^n_+\,.
\end{displaymath}
Theorem \ref{thm5} implies that $\pi^2 I - A(\|u\|_{\infty})$ is
$\mathbb{R}^n_+$-monotone. Then using Theorem \ref{thm4} we obtain
\begin{equation}
(\pi^2 I - A(\|u\|_{\infty}))^{- 1}(s + b) - r  \in
\mathbb{R}^n_+\,, \label{eq:eq13}
\end{equation}
which implies that
\begin{displaymath}
\pi\|(\pi^2 I - A(\|u\|_{\infty}))^{- 1}\|(\|S(u)\|_2 + \|b\|)
\geq \|u'_j\|_2 \,,
\end{displaymath}
for $j = 1, \dots , n$. Using Lemma \ref{lem1} and the fact that
$\|S(u)\|_2 \leq \|S(u)\|_{\infty}$ we deduce that
\begin{equation}
\|S(u)\|_{\infty} \geq \frac{2\|u\|_{\infty}}{\pi\|(\pi^2 I -
A(\|u\|_{\infty}))^{- 1}\|} - \|b\|\,. \label{eq:eq14}
\end{equation}
We shall prove that \eqref{eq:eq14} implies that
$S: (X, N_1) \to (Y, \|.\|_{\infty})$ is a proper map.
Let $(u_n)_{n \in \mathbb{N}}$ be a sequence in $X$ and
$v \in Y$ such that $S(u_n) \to v$ as
$n \to + \infty$. \eqref{eq:eq8} and \eqref{eq:eq14} imply that
there exists a constant $M > 0$ such that
$\|u_n\|_{\infty} \leq M$ for every $n \in \mathbb{N}$.
Since $K: (X, N_1) \to (Y, \|.\|_{\infty})$ is a compact operator,
it follows that the sequence $(K(u_n))_{n \in \mathbb{N}}$ contains
a convergent subsequence. Without loss of generality we may assume that
$(K(u_n))_{n \in \mathbb{N}}$ is convergent to $w \in Y$. Letting
$n \to + \infty$ in the equality
\begin{displaymath}
u_n = L^{- 1}S(u_n) + L^{- 1}K(u_n) \,,
\end{displaymath}
we obtain
\begin{equation}
\lim_{n \to +\infty}\|u_n - L^{- 1}(v) - L^{- 1}(w)\|_{\infty} =
0\,. \label{eq:eq15}
\end{equation}
Then we have
\begin{align*}
&\lim_{n \to +\infty}\|L(u_n) - L(L^{- 1}(v) + L^{-1}(w))\|_{\infty}\\
&= \lim_{n \to +\infty}\|(S(u_n) - v) + (K(u_n) - w)\|_{\infty} = 0\,.
%\label{eq:eq16}
\end{align*}
  From this equality and \eqref{eq:eq15}, we deduce that
\begin{displaymath}
\lim_{n \to +\infty}N_1(u_n - L^{- 1}(v + w)) = 0\,.
\end{displaymath}
Therefore $S: (X, N_1) \to (Y, \|\cdot\|_{\infty})$ is a proper map.
Using Theorem \ref{thm7} we conclude that $S$ is a global homeomorphism and
consequently the equation $Su = 0$ has a unique solution $u \in
X$. This is also the unique solution of the boundary value problem
\eqref{eq:eq1}.

\section{Examples}

In this section we give two examples to illustrate Theorems
\ref{thm2} and \ref{thm3}.
Define $a$, $h: \mathbb{R} \to \mathbb{R}_+$ by
\begin{displaymath}
a(t) = \begin{cases}
0 & \text{if }  t \leq 1\,,\\
  1 - \frac{1}{t^{\alpha}} & \text{if } t \geq 1
\end{cases}\quad \text{and} \quad
  h(t) = \int_1^t{a(s)\, ds}\,, \quad t \in \mathbb{R} \,,
\end{displaymath}
where $\alpha > 0$.

\begin{example} \label{exa1} \rm
  Let $n =2$. We set
\begin{displaymath}
f_1(x,u) = \pi^2 h(u_1) + g_1(x),\quad
f_2(x,u) = |u_1|^{\beta} + \pi^2h(u_2) + g_2(x)\,,
\end{displaymath}
for $(x,u) \in [0,1]\times\mathbb{R}^2$. $\beta > 1$ is a constant
and $g_1$, $g_2 \in C([0,1],\mathbb{R})$. Then we can take
\begin{gather*}
a_{11} = a_{22} = \pi^2 a \,, \quad
a_{12} = 0 \,, \quad
a_{21}(t) = t^{\beta - 1} \,, \quad t \geq 0 \,,\\
b_j = \|g_j\|_{\infty} \,,\quad j = 1, \, 2 \,,
\\
\lambda_{11} = \lambda_{22} = \pi^2 a \,, \quad
\lambda_{12} = 0 \,, \quad
\lambda_{21}(t) = \beta t^{\beta - 1} ,\quad  t \geq 0\,.
\end{gather*}
   We easily verify that $\rho(A(t)) = \rho(\Lambda(t)) = \pi^2 a(t) <
\pi^2$ for $t\geq 0$,
\begin{gather*}
\frac{t}{\|(\pi^2 I - A(t))^{-1}\|} = \frac{\pi^4
t^{1-\alpha}}{\pi^2 + t^{\alpha + \beta - 1}} \quad \textrm{for}
\quad t \geq 1 \,,
\\
\|(\pi^2 I - \Lambda(t))^{- 1}\| = \frac{t^{\alpha}}{\pi^2} +
\frac{\beta}{\pi^4}t^{2\alpha + \beta - 1} \quad \textrm{for} \;
t\geq 1\,.
\end{gather*}
Note that $a_{21}$ and $\lambda_{21}$ are unbounded.
If $2\alpha + \beta < 2$ we can use either Theorem \ref{thm2} or
Theorem \ref{thm3}.
Now let $2\alpha < 1$ and
$\beta = 2(1 - \alpha)$. Then Theorem \ref{thm2}
applies but Theorem \ref{thm3} does not apply.
\end{example}

\begin{example} \label{exa2} \rm
  Let $n =2$. We set
\begin{displaymath}
f_1(x,u) = \pi^2h(u_1) + g_1(x), \quad
f_2(x,u) = \cos|u_1|^{\beta} + \pi^2h(u_2) + g_2(x)\,,
\end{displaymath}
for $(x,u) \in [0,1]\times\mathbb{R}^2$. $\beta > 1$ is a constant
and $g_1$, $g_2 \in C([0,1],\mathbb{R})$. Then we can take
\begin{gather*}
a_{11} = a_{22} = \pi^2 a \,, \quad
a_{12} = a_{21} = 0 \,, \quad
b_1 = \|g_1\|_{\infty} \,, \quad
b_2 = 1 + \|g_2\|_{\infty} \,, \\
\lambda_{11} = \lambda_{22} = \pi^2 a \,, \quad
\lambda_{12} = 0 \,, \quad
\lambda_{21}(t) = \beta t^{\beta - 1}  ,\quad  t \geq 0\,.
\end{gather*}
   We easily verify that
$\rho(A(t)) = \rho(\Lambda(t)) = \pi^2 a(t) <\pi^2$ for $t\geq 0$,
\begin{gather*}
\frac{t}{\|(\pi^2 I - A(t))^{-1}\|} = \frac{ t^{1-\alpha}}{\pi^2}
\quad \text{for } t \geq 1 \,,
\\
\|(\pi^2 I - \Lambda(t))^{- 1}\| = \frac{t^{\alpha}}{\pi^2} +
\frac{\beta}{\pi^4}t^{2\alpha + \beta - 1} \quad \textrm{for} \;
t\geq 1\,.
\end{gather*}
Notice that $\lambda_{21}$ is unbounded. If
$2\alpha + \beta \leq 2$, then Theorem \ref{thm2} and Theorem 
\ref{thm3} apply. If
$2\alpha + \beta > 2$ and $\alpha < 1$, Theorem \ref{thm3} still applies
but not Theorem \ref{thm2}.
\end{example}

We conclude this paper with the following remark.

\begin{remark} \label{rmk2} \rm
  With the notations of Theorems \ref{thm2} and \ref{thm3},
assume that $\lambda_{jk}$ are bounded for $j$, $k = 1, \dots, n$.
Then \eqref{eq:eq4} implies \eqref{eq:eq5}. Indeed we have
\begin{displaymath}
(\pi^2 I - \Lambda(t))^{- 1} = \frac{1}{\det(\pi^2 I -
\Lambda(t))}B(t) ,\quad t \geq 0
\,,
\end{displaymath}
where $B(t) \in M(n)$ is nonnegative and
$\det(\pi^2 I - \Lambda(t))>  0$ (see Remark \ref{rmk1}).
Since $\lambda_{jk}$ are bounded for $j$, $k = 1, \dots, n$, there
exists a constant $d > 0$ such that
$\|B(t)\| \leq d $ for all $t \geq 0$.
Then we can write
\begin{displaymath}
\frac{1}{\|(\pi^2 I - \Lambda(t))^{- 1}\|}
= \frac{\det(\pi^2 I - \Lambda(t))}{\|B(t)\|} \geq \frac{1}{d}
\det(\pi^2 I - \Lambda(t))\,, \quad t \geq 0 \,,
\end{displaymath}
and our claim follows.
\end{remark}

It is easily seen that \eqref{eq:eq5} does not imply \eqref{eq:eq4}
in general. Indeed let
\[
\lambda_{11}(t) = \lambda_{22}(t) = \pi^2(1 - \frac{1}{t}) \,,
\quad t \geq 1
\]
and $\lambda_{12} = \lambda_{21} = 0$.
Then we have
\begin{displaymath}
\frac{1}{\|(\pi^2 I - \Lambda(t))^{- 1}\|} = \frac{\pi^2}{t} \quad
\textrm{and} \quad
\det(\pi^2 I - \Lambda(t)) = \frac{\pi^4}{t^2}\,, \quad  t \geq 1 \,.
\end{displaymath}

\begin{thebibliography}{0}
\bibitem{bepl} A. Berman and J. Plemmons;
\emph{Nonnegative matrices in the mathematical sciences},
Academic Press, New York, 1979.

\bibitem{bm} S. Banach and S. Mazur;
\emph{\"Uber mehrdeutige stetige Abbildungen},
Studia Math. 5 51934, 174-178.

\bibitem{ca} R. Caccioppoli;
\emph{sugli elementi uniti delle
transformazioni funzionali: un teorema di esistenza e di
unicita ed alcune sue applicazioni}, Rend. Seminar Mat.
Padova, 3 (1932), 1-15.

\bibitem{da} R. Dalmasso;
\emph{An existence and uniqueness
theorem for a second order nonlinear system}, Journal
of Mathematical Analysis and Applications 327 (2007),
715-722.

\bibitem{ra1} M. R\v adulescu and S. R\v adulescu;
\emph{An application of a global inversion theorem to a Dirichlet
problem for a second order differential equation},
Rev. Roumaine Math. Pures Appl. 37 (1992), 929--933.

\bibitem{ra2} M. R\v adulescu and S. R\v adulescu;
\emph{Applications of a global inversion theorem to unique solvability of
second order Dirichlet problems}, Annals of University of Craiova,
Math. Comp. Sci. Ser. 30 (1) (2003), 198-203.

\end{thebibliography}

\end{document}