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\markboth{\hfil Resonance with respect to the Fu\v c\'\i k spectrum
\hfil EJDE--2000/37}{EJDE--2000/37\hfil
A. K. Ben-Naoum, C. Fabry, \& D. Smets \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~37, pp.~1--21. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
  \vspace{\bigskipamount} \\
%
   Resonance with respect to the Fu\v c\'\i k spectrum
\thanks{ {\em Mathematics Subject Classifications:} 70K30, 35P30, 35G30.
\hfil\break\indent
{\em Key words:} Resonance, jumping nonlinearity, Landesman-Lazer conditions.
\hfil\break\indent
\copyright 2000 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted January 4, 2000. Published May 17, 2000.
\hfil\break\indent
(D. S.) Supported by the Fonds National de la Recherche Scientifique
} }
\date{}
%
\author{ A. K. Ben-Naoum, C. Fabry, \& D. Smets }
\maketitle

\begin{abstract}
Let $L$ be a self-adjoint operator on $L^2(\Omega;\mathbb R)$ with 
$\Omega$ a bounded and
open subset of ${\mathbb R}^N$. This article considers the resonance problem
  with respect to the  Fu\v c\'\i k spectrum of $L$, which means that 
we study equations of the form
$$
   Lu = \alpha u^+ - \beta u^- + f(\cdot,u),
$$
when the homogeneous equation  $Lu = \alpha u^+ - \beta u^-$
has  non-trivial solutions. Using the computation of degrees that are
not necessarily $+1$ or $-1$, we present results about the
existence of solutions. Our results are illustrated
with examples and can be seen as generalizations of Landesman-Lazer
conditions. Non-existence results are also given.
\end{abstract}


\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\newtheorem{cor}{Corollary}


\section{Introduction}
In this paper, we continue the study started in \cite{bfs} about
nonlinear equations containing an asymmetric nonlinear term
(or ``jumping nonlinearity''); i.e., equations of the form
\begin{equation}
Lu = \alpha u^+ - \beta u^- + f \label{eq:1}
\end{equation}
or, more generally,  of the form
\begin{equation}
Lu = \alpha u^+ - \beta u^- + f(\cdot,u), \label{eq:1b}
\end{equation}
where $u^+ = \max \{u,0\}$, and $u^- = \max \{-u,0 \}$.
The linear operator $L$ is defined from
$\mathop{\rm dom} L \subset L^2 (\Omega; \mathbb{ R })$ to
$L^2 (\Omega; \mathbb{ R })$, where
$\Omega$ is a bounded open subset of ${\mathbb R}^N$.
On this operator we shall assume the hypothesis

\begin{description}

\item{(H1)} The operator $L$ is self-adjoint and has and eigenvalue
$\lambda^*$ such that $\dim \ker (L - \lambda^* I)=n  <  \infty$.
\end{description}
For studying  (\ref{eq:1b}), we assume that $L$ has a
compact resolvent and that $f$ is (globally)
bounded by an $L^2$-function. The precise hypotheses about
$f(t,u)$ will be stated in Section 4.
Also we assume that the pair $(\alpha,\beta)$ is ``not too far'' from
$(\lambda^{\ast},\lambda^{\ast} )$, in the sense of the following
hypothesis, in which $\sigma(L)$ denotes the spectrum of $L$.
\begin{description}

\item{(H2)} The values $\alpha$ and $\beta$ are in a closed interval $I$ such
that $I \cap \sigma (L) = \{ \lambda^{\ast} \}$.
\end{description}

The set of points at which the homogeneous equation
\begin{equation}
Lu = \alpha u^+ - \beta u^- \label{eq:2}
\end{equation}
has nontrivial solutions is called the Fu\v c\'\i k or
Dancer-Fu\v c\'\i k spectrum, and is denoted by $\Sigma(L)$.
Dancer \cite{dan} and Fu\v c\'\i k \cite{fu}
recognized the importance of this set in
the study of semi-linear boundary value problems.
Under hypotheses which include (H1) and (H2) as a particular case,
we have studied  the structure of the
Fu\v c\'\i k spectrum within $I\times I$, \cite{bfs}.  We also showed the
existence of solutions for (\ref{eq:1}) and (\ref{eq:1b}) when
$(\alpha,\beta)$ does not belong to the Fu\v c\'\i k spectrum and when
$f(t,u) $ has sublinear growth in $u$ as  $|u| \to \infty$.

In the present work, we investigate the case when $(\alpha,\beta)$ does
belong to the Fu\v c\'\i k spectrum. This situation can be considered as
a situation of resonance and is divided into two different cases.

In Section \ref{section:first}, we obtain, for fixed $(\alpha,\beta)$ in
$\Sigma(L) \cap (I \times I),$ existence results for (\ref{eq:1})
under some regularity assumption. Roughly, the assumption means that the set
$\left\{\varphi \mid  L\varphi = \alpha \varphi^+-\beta \varphi^-\right\}
\setminus \{0\}$ is locally a manifold  whose dimension may differ
on each component, but its tangent space at $\varphi$ is exactly
$\ker(L-\alpha \chi(\varphi^+)I-\beta \chi(\varphi^-)I)$.  Here,
$\chi({\varphi^+})$ is the characteristic function of the set
$\{ t \in \Omega \mid \varphi(t) > 0 \}$, and $\chi({\varphi^-  })$ is
defined similarly. Under this regularity assumption, a Landesman-Lazer type
condition is obtained for showing the
existence of at least one solution to (\ref{eq:1}).
The regularity assumption is satisfied in particular when all solutions of
(\ref{eq:2}) are positive multiples of a finite number of particular
solutions, and
$$
\mathop{\rm dim} \ker(L-\alpha \chi(\varphi^+)I-\beta \chi(\varphi^-)I)=1
$$
for any such nontrivial solution.
The regularity assumption is also satisfied when
\begin{description}

\item{(H3)} For each $x \in \ker (L - \lambda^{\ast}
I)$ there exists a solution $\varphi_x$
of (\ref{eq:2}), such that $P \varphi_x =x$,
where $P$ denotes the orthogonal projection onto
$\ker (L- \lambda^{\ast} I)$.
\end{description}

However, under (H3), better existence results, still based on
topological degree arguments, can be
obtained, which also apply to equation (\ref{eq:1b}). This is the object
  of Section \ref{section:H3}.  It corresponds to a Fu\v c\'\i k
  spectrum in $I\times I $ reduced to a curve. The existence of solutions
is established if a certain degree is not equal to zero.
Consider the particular case where $f$ has limits
$f_+(t) $ and $f_-(t)$  for
$u \to \pm \infty$. Then, it will be shown that the
degree in question can be computed from the indices of the critical
values of the function
$$ \Psi :S^{n-1} \to {\mathbb R}, \quad  x \mapsto \langle f_+, 
\varphi_x^+ \rangle
-  \langle f_-,\varphi_x^- \rangle,
$$
where $S^{n-1}$ is the unit sphere in $\ker(L-\lambda^*I)$ and
$\langle\,\cdot\,,\, \cdot \,\rangle $
denotes the scalar product in $L^2 (\Omega; \mathbb{ R })$
(the norm will be denoted by $\| \cdot\|)$. When
$\mathop{\rm dim}\ker(L-\lambda^* I)=2$, the existence result
takes a particularly simple form. Let
$\{ v^{(1)}, v^{(2)} \}$ denote an orthonormal basis of
$\ker (L - \lambda^{\ast} I)$, and let
$$
z_{\theta} = \cos \theta \,v^{(1)} + \sin \theta \,v^{(2)}\,.
$$
We will abbreviate notation by using $\varphi_{\theta}$ for
$\varphi_{z_{\theta}}$.
Under appropriate hypotheses on $L,\alpha,\beta$ (see Corollary
\ref{theo:5b} and Theorem \ref{theo:degree}), if the  function
$\Psi : \theta \mapsto \langle f_+,\varphi^+ _{\theta} \rangle - \langle
f_-,\varphi^-_{\theta} \rangle$ has
only simple zeros, and if the number of zeros in $[0,2 \pi)$ is not 2,
equation (\ref{eq:1b}) has at least one solution.

The results obtained in
Section \ref{section:H3} can be seen as generalizations of
the Landesman-Lazer conditions. Those conditions (see
\cite{maw2} for a survey) are familiar in the context of
resonance with respect to the ``usual spectrum''
(i.e., when $\alpha=\beta$) and correspond to a function
$\Psi$ of constant sign. It must be noticed that, even when
$\alpha=\beta$, the conditions presented in
Theorem \ref{theo:degree} and some of its corollaries provide a
generalization of  the classical Landesman-Lazer
conditions. The existence conditions of Section \ref{section:H3}
have been inspired by results of \cite{fa-fo} concerning the
periodic boundary-value problem for the second order equation
$$ u'' + \alpha u^+ -\beta u^-= f(t)\,.$$

Section \ref{section:non} concludes our study with a non-existence result
for equation (\ref{eq:1}). Now we present a few results needed later.

\section{Preliminary results}\label{section:prelim}

Let $f_+$ and $f_- $ belong to $L^2 (\Omega; \mathbb{ R })$, with
  $f_+-f_- \in L^{\infty} (\Omega; \mathbb{ R })$.
For $\varepsilon \geq 0$, define
\begin{equation}
f_{\varepsilon} (t,u)= \frac{1}{2}\, \left[f_+(t) +f_-(t)\right] +
\frac{1}{\pi}\, \left[f_+(t) -f_-(t)\right] \arctan(\varepsilon
u)\,.
\end{equation}
Clearly, $f_{\varepsilon}(t,u)$ is Lipschitz continuous in
$u$, with Lipschitz constant $
\frac{\varepsilon }{\pi} \, |f_+(t) -f_-(t)|$. To
$f_{\varepsilon}$, we associate the mapping
\begin{equation}
N_{\varepsilon} : L^2 (\Omega; \mathbb{ R }) \rightarrow L^2
(\Omega; \mathbb{ R }):u \mapsto \alpha u^+ -\beta u^-
+f_{\varepsilon}(\cdot,u)\,.
\label{eq:nne2}
\end{equation}
It is easy to show that $N_{\varepsilon}-\frac{1}{2}(\alpha+\beta) I$ is
Lipschitz continuous
with Lipschitz constant $\frac{1}{2}|\alpha-\beta| + \frac{\varepsilon
}{\pi}\|f_+-f_-\|_{L^{\infty}}$.
The following lemma is a slight generalization of Lemma 1 in \cite{bfs},
to which we refer for the proof.

\begin{lemma}  \label{lemma:1} Let $L, \alpha, \beta$ satisfy
(H1) and (H2), and $f_{\varepsilon}$ be as above.  Then, provided that
$\varepsilon$ is small enough, for any
$x \in \ker (L - \lambda^{\ast} I)$, the problem
\begin{eqnarray}
&Lu = \lambda^{\ast} x  + (I - P) [\alpha u^+ - \beta u^- +
f_{\varepsilon}(\cdot,u)], &\label{eq:3a}\\
&Pu = x &\label{eq:3b}
\end{eqnarray}
has a unique solution $u_x = u_x (f_{\varepsilon}, \alpha, \beta)$.
Moreover, this solution is Lipschitzian with respect to
$x,\alpha,\beta$.  More precisely, for $x,y$ in a bounded set
$B_1 \subset\ker (L
- \lambda^{\ast} I)$, there exists a constant $K>0$ such that
$$
\|u_x (f_{\varepsilon},\alpha,\beta) - u_y
(f_{\varepsilon},\alpha',\beta')\| \leq K \left(
\|x-y\| + |\alpha-\alpha' | + |\beta-\beta'|\right).
$$
\end{lemma}

Let $u_x = u_x (f_{\varepsilon},\alpha,\beta)$ be the solution of
(\ref{eq:3a}), (\ref{eq:3b}).  Define
\begin{eqnarray*}
c (x,f_{\varepsilon},\alpha,\beta) &=& - P [\alpha u^+_x -\beta
u^-_x + f_{\varepsilon}(\cdot,u_x)] + \lambda^{\ast} x ,
\end{eqnarray*}
so that $u_x $ satisfies
\begin{equation}
Lu_x = \alpha u^+_x - \beta u^-_x +  f_{\varepsilon}(\cdot,u_x) + c
(x,f_{\varepsilon},\alpha,\beta)  .
\label{eq:6}
\end{equation}
It is clear that
\begin{equation}
Lu = \alpha u^+ - \beta u^- + f_{\varepsilon}(\cdot,u) \label{eq:6b}
\end{equation}
  admits a solution if and only if there exists $x \in
\ker(L-\lambda^* I)$ such that  \\ $c (x,f_{\varepsilon},\alpha,\beta) = 0$.
Therefore, (\ref{eq:6b}) can be reduced to a problem in a
finite-dimensional space. In relation with equation (\ref{eq:2}), we will
write $c_0 (x,\alpha,\beta)$ for
$c(x,0,\alpha,\beta)$.   Hence, we have
$$c_0 (x,\alpha,\beta) = - P [\alpha u^+_x - \beta u^-_x ] +
\lambda^{\ast} x,$$
where $u_x =u_x(0,\alpha,\beta)$, given by Lemma 1, is the unique solution of
\begin{eqnarray*}
&Lu =\lambda^{\ast} x  + (I - P) [\alpha u^+ - \beta u^-] ,&\\
&Pu = x\,.&
\end{eqnarray*}
To the function $c_0$, we associate the function
$$h_0 : \ker(L-\lambda^* I) \times I \times I : (x,\alpha,\beta) \mapsto
\langle c_0(x,\alpha,\beta), x\rangle\,;
$$
some of its properties, established in \cite{bfs}, are recalled below.

\begin{lemma} \label{lemma:x3} Let (H1) and (H2) hold.
Then, the function $h_0$ admits partial derivatives with respect to
$\alpha,\beta \in I$, is differentiable with respect to $x \in
\ker(L-\lambda^* I) $ ,
$$ \frac{\partial}{\partial \alpha} h_0 (x,\alpha,\beta)= -
\| u^+_x \|^2,\quad  \frac{\partial}{\partial \beta} h_0 (x,\alpha,\beta)= -
\| u^-_x \|^2 ,
$$
and
\begin{equation}
\nabla_x h_0 (x,\alpha,\beta) = 2 c_0(x,\alpha,\beta).
\label{eq:grad}
\end{equation}
\end{lemma}

Based on the function $h_0$, the following theorem, which has
been proved in \cite{bfs} and in \cite{mag}, provides a
characterization of the points of the Fu\v c\'\i k spectrum, within the
square $I\times I$.

\begin{theorem}  \label{theo:1} Let (H1) and (H2) hold.
Then, $(\alpha, \beta)$ in $I \times I$ belongs to the Fu\v c\'\i k spectrum
of $L$ if and only if $0$ is a
critical value of the function
$$h_0(\cdot,\alpha,\beta) : x \mapsto
\langle  c_0 (x,\alpha,\beta), x \rangle, $$
with the critical value being reached at some point $x \neq 0$.
\end{theorem}


The Fu\v c\'\i k spectrum contains in particular the sets
$$F^- = \{ (\alpha,\beta) \in I \times I\mid \min_{x \, \in \,\ker (L -
\lambda^{\ast} I), \,\|x\| =1} \langle
c_0 (x,\alpha,\beta),x \rangle  = 0 \}$$
and
$$F^+ = \{ (\alpha,\beta) \in I \times I\mid \max_{x \,\in \,\ker (L -
\lambda^{\ast} I), \,\|x\| =1} \langle c_0
(x,\alpha,\beta),x \rangle  = 0 \}.$$


\section{A first existence result at resonance} \label{section:first}

Let $(\alpha,\beta)$ be fixed in $\Sigma(L) \cap I \times I$. We will obtain a
Landesman-Lazer type existence condition for equation (\ref{eq:1})
under a regularity assumption.  Let  $\mathcal{S}$ denote the set of
all solutions of equation (\ref{eq:2}), and assume that
\begin{description}

\item{(H3')}  For each $\varphi \in \mathcal{S},\, \varphi \ne 0$, there
exists $\delta>0$ such that, if $\tilde{\varphi} \in \mathcal{S}$ and
$\|\varphi-\tilde{\varphi}\| \le \delta$, then
$\mathrm{dim}\ker(L-\alpha \chi(\varphi^+)I - \beta
\chi(\varphi^-)I)=\mathrm{dim}\ker(L-\alpha \chi(\tilde{\varphi}^+)I - \beta
\chi(\tilde{\varphi}^-)I)$.
Moreover, for each $\psi \in
\ker(L-\alpha \chi(\varphi^+)I - \beta \chi(\varphi^-)I)$,
$\,\textrm{dist}(\varphi+\varepsilon \psi, \mathcal{S})=o(\varepsilon)\:$ for
$\varepsilon \to 0$.
\end{description}

If it happens that $\mathcal{S}$ is locally a manifold whose tangent
  space at $\varphi$ is exactly $\ker(L-\alpha \chi(\varphi^+)I-\beta
\chi(\varphi^-)I)$, then (H3') is clearly satisfied. This will be the
case when all solutions of (\ref{eq:2}) are positive multiples of a
finite number of particular solutions and
$$ \mathop{\rm dim} \ker(L-\alpha \chi(\varphi^+)I-\beta \chi(\varphi^-)I)=1,
$$
for any such nontrivial solution $\varphi$.  The condition (H3')
also holds when (H3) is satisfied; but, in
this last case, better existence results can be obtained, as shown in
Section \ref{section:H3}.

For the sake of simplicity, in this section we will  consider only
equation (\ref{eq:1}) with a function $f $ independent of the unknown
function $u$.
We will also need the following (weak) hypotheses
\begin{description}

\item{(H4)} For some $p>2$,  $\mathop{\rm dom}L$, equipped with the graph
norm, is continuously injected into
$L^p(\Omega;\mathbb R)$

\item{(H5)} For any nontrivial solution $\varphi$ of the homogeneous
equation (\ref{eq:2}),
$$\mathop{\rm meas}\{t \in \Omega \mid \varphi(t) = 0\}=0.$$
\end{description}


\begin{theorem} \label{theo:5}
Assume that $(\alpha,\beta)$ belongs to the Fu\v c\'\i k spectrum,
that (H1), (H2), (H3'), (H4), and (H5) hold, that the
homogeneous equation (\ref{eq:2}) has no solution of
constant sign, and that for any nontrivial solution $\varphi$ of
(\ref{eq:2}), we have $\langle f,\varphi \rangle >0$.
Moreover, assume that, for $\varepsilon>0$ small enough,
$(\alpha +\varepsilon, \beta +\varepsilon)$ does not belong to the
Fu\v c\'\i k spectrum and the degree, with respect to open bounded
sets containing $0$, of the mapping
$c_0(\cdot ,\alpha +\varepsilon, \beta
+\varepsilon):\ker(L-\lambda^* I) \to
\ker(L-\lambda^* I) $, is different from $0$.
Then, equation (\ref{eq:1}) has at least one solution.
\end{theorem}

The sign condition can be reversed to $\langle f,\varphi \rangle <0$,
taking then $\varepsilon<0$ in the
condition on the degree.


\paragraph{Proof.} We will use the homotopy
\begin{equation}
  c(x,f,\alpha + \varepsilon s,\beta +\varepsilon s)=0\,, \label{eq:ho}
\end{equation}
where $s \in [0,1]$. By the last hypothesis, for $\eta $ small enough,
the degree, with respect to the unit ball centered at $0$, of the mapping
$x \mapsto c(x,\eta f,\alpha+\varepsilon,\beta+\varepsilon)$ is
different from $0$ and, since
\begin{equation}
c(rx,rf,\alpha',\beta')=r
\,c(x,f,\alpha',\beta') \mbox{ for } r>0 \mbox{ and for any }\alpha',\beta'
\in I, \label{eq:14b}
\end{equation}
the same is true for the degree of
$c(x,f,\alpha+\varepsilon,\beta+\varepsilon)$, with respect to sufficiently
large
balls centered at $0$. Using the invariance of the degree with respect to a
homotopy, the theorem is then proved if we can find a priori bounds for the
solutions of (\ref{eq:ho}). By contradiction, assume that there exists
sequences $\{s_n\}, \{x_n\}$,  with $s_n \in [0,1]$, $\{x_n\} \subset
\ker(L-\lambda^* I)$,
$\|x_n\| \to \infty$, such that
\begin{equation}
c(x_n,f,\alpha + \varepsilon s_n,\beta + \varepsilon s_n)=0 \;. \label{eq:deg1}
\end{equation}
Denoting by $u_n$ the function associated to $x_n$ by Lemma \ref{lemma:1}
$(u_n=u_{x_n}(f,\alpha,\beta))$, we can equivalently write
\begin{equation}
Lu_n= (\alpha + \varepsilon s_n) u_n^+ - (\beta + \varepsilon s_n) u_n^- +
f \;.
\label{eq:deg1b}
\end{equation}
Let $y_n=x_n/\|x_n\|\;;$ using (\ref{eq:14b}) again, we have
\begin{equation}
c\left(y_n,\frac{f}{\|x_n\|},\alpha + \varepsilon
s_n,\beta +\varepsilon s_n\right)=0\,.\label{eq:lim}
\end{equation}
Passing, if necessary, to subsequences, we can assume that $\{y_n\}, \{s_n\},
$ converge; let their respective limits be denoted by
$y^*,s^*$. Going to the limit in (\ref{eq:lim}), we obtain, $c$ being
continuous,
\begin{equation}
c_0(y^*,\alpha +\varepsilon s^*,\beta +\varepsilon s^*)=0\,.\label{eq:c00}
\end{equation}
Since $\|y^*\|=1$, this implies that $(\alpha +\varepsilon s^*,\beta
+\varepsilon
s^*)$ belongs to the Fu\v c\'\i k spectrum, from which follows, by the last
hypothesis of the theorem, that
$s^*=0. $ Consequently, the function $u_{y^*}(0,\alpha,\beta)$, associated to
$y^*$ by Lemma \ref{lemma:1}, is a solution of (\ref{eq:2}).
In the sequel, we will write $\varphi$ for $u_{y*}(0,\alpha,\beta). $
Let $v_n=u_{n}/\|x_n\|$, and $\varphi_n \in \mathcal{S}$ be such that
$$
\|\varphi_n-v_n\| = \mathop{\rm dist}(v_n,\mathcal{S}).
$$
Hence, both $\{v_n\}$  and $\{\varphi_n\}$ converge to $\varphi$.
By (\ref{eq:deg1b}),  the equation verified by $v_n$ can be written
under the form
\begin{eqnarray}
Lv_n&=& \alpha \chi(\varphi_n^+)v_n + \beta \chi(\varphi_n^-) v_n  \nonumber\\
&&+ \varepsilon s_n \chi(\varphi_n^+) v_n + (\alpha + \varepsilon
s_n)[\chi(v_n^+) - \chi(\varphi_n^+)] v_n  \label{eq:ll1} \\
&& + \varepsilon s_n \chi(\varphi_n^-) v_n + (\beta + \varepsilon
s_n)[\chi(v_n^-) - \chi(\varphi_n^-)] v_n + \frac{f}{\|x_n\|} \,. \nonumber
\end{eqnarray}
Since $\varphi_n \in \ker (L -\alpha \chi(\varphi_n^+)I -\beta
\chi(\varphi_n^-)
I), $ equation (\ref{eq:ll1}) has a solution only if
\begin{eqnarray}
\varepsilon s_n \langle \chi(\varphi_n^+) v_n, \varphi_n \rangle
+ (\alpha + \varepsilon s_n)\langle [\chi(v_n^+)
- \chi(\varphi_n^+)] v_n, \varphi_n \rangle
&& \label{eq:fredh}\\
+ \varepsilon s_n \langle\chi(\varphi_n^-) v_n, \varphi_n \rangle + (\beta +
\varepsilon s_n)\langle [\chi(v_n^-) - \chi(\varphi_n^-)] v_n,
\varphi_n \rangle +
\frac{\langle f,  \varphi_n \rangle}{\|x_n\|} &=&0\,. \nonumber
\end{eqnarray}
Let $w_n$ denote the orthogonal projection of $v_n$ onto the space
$K_n:=\ker(L-\alpha \chi(\varphi_n^+)I-\beta \chi(\varphi_n^-)I)$.  If
$w_n \ne \varphi_n$, then by assumption (H3'),
$$
\mathop{\rm dist}(v_n,\mathcal{S})\leq
\mathop{\rm dist}(v_n,\varphi_n+\varepsilon(w_n-\varphi_n))+o(\varepsilon)
= \mathop{\rm dist}(v_n,\varphi_n)-O(\varepsilon)+o(\varepsilon),
$$
contradicting the definition of $\varphi_n$.  Hence, $\varphi_n=w_n$ and
\begin{equation}
L\varphi_n= \alpha \chi(\varphi_n^+)\varphi_n + \beta
\chi(\varphi_n^-) \varphi_n.\label{eq:wn}
\end{equation}
By the arguments used to prove Lemma \ref{lemma:1}, it can be shown that, under
hypotheses
(H1) and (H2), the operator $[L -\alpha \chi(\varphi_n^+)I -\beta
\chi(\varphi_n^-)I
]_{\varphi_n^{\perp}} : K_n^{\perp} \to K_n^{\perp}$ is invertible.
Using the norm of its inverse and the fact that $\mathop{\rm dom}L$,
equipped with the graph norm, is continuously injected into
$L^p(\Omega;\mathbb R)$, we can show, subtracting (\ref{eq:wn})
from (\ref{eq:ll1}) and using the fact that $\varphi_n -v_n \in K_n^{\perp}$,
that
\begin{eqnarray}
\lefteqn{\|v_n -\varphi_n\|_{L^p}  } \label{eq:bd1} \\
&\leq& C_n \Bigl[ \varepsilon s_n +\frac{\|f\|}{\|x_n\|} +
\|[\chi(v_n^+)  -\chi(\varphi_n^+)] v_n \|
+ \|[\chi(v_n^-) -\chi(\varphi_n^-)] v_n \| \Bigr]\,, \nonumber
\end{eqnarray}
for some constant $C_n$.  By assumption (H3'), and the
semi-continuity of separated parts of the spectrum (see Theorem 4.3.16 in
Kato \cite{Kato}), $C:=\sup_{n \in \mathbb{N}}C_n < +\infty$.
But, for $n$ large,
\begin{equation}
\|[\chi(v_n^+) -
\chi(\varphi_n^+) ]v_n \| ^2 = \int_{v_n \varphi_n <0} v_n^2
\leq  \int_{v_n \varphi_n <0} (v_n-\varphi_n) ^2\,. \label{eq:vw}
\end{equation}
Using H\"older's inequality and the fact that $\mathop{\rm dom}L 
\subset L^p(\Omega;
\mathbb R)$ for some $p>2$,
we can write
\begin{equation}
\int_{v_n \varphi_n <0} (v_n-\varphi_n)^2 \leq
\|v_n-\varphi_n\|^2_{L^p}
\mathop{\rm meas}\big\{t\in\Omega\mid v_n(t) \varphi_n(t) <0\big\}^{1-2/p}.
\label{eq:vw2}
\end{equation}
As both $\{v_n\}$, $\{\varphi_n\}$ converge to $\varphi$ in $L^2 (\Omega;
\mathbb{ R })$, we have,
by hypothesis (H5),
$$ \mathop{\rm meas}\{ v_n(t)\varphi_n(t) <0 \}\le
\mathop{\rm meas}\{v_n(t) \varphi(t)<0 \}+\mathop{\rm meas}\{
\varphi_n(t)\varphi(t)<0 \} \stackrel{n \to
\infty}{\to} 0 \,.
$$
Consequently, by (\ref{eq:vw}) and (\ref{eq:vw2}),
$\|(\chi(v_n^+)-\chi(\varphi_n^+))v_n\|=o(\|v_n-\varphi_n\|_{L^p})
$ for $n \to \infty$.
A similar result holds for $\|(\chi(v_n^-) -
\chi(\varphi_n^-)) v_n \|$. Hence, for $n$ large, the last two terms in
(\ref{eq:bd1}) can be  combined with the left hand side to yield an inequality
of the type
\begin{equation}
\|v_n -\varphi_n\|_{L^p} \leq C'\big[ \varepsilon s_n + \frac{\|f\|}{\|x_n\|}
\big] \,,\label{eq:bd2}
\end{equation}
for some $C'>C$. Taking (\ref{eq:vw}), (\ref{eq:vw2}) into account, we then
see that, in
(\ref{eq:fredh}), the terms $(\alpha + \varepsilon
s_n)\langle [\chi(v_n^+) - \chi(\varphi_n^+)] v_n, \varphi_n \rangle$ and
$(\beta +
\varepsilon s_n)\langle [\chi(v_n^-) - \chi(\varphi_n^-)] v_n, \varphi_n
\rangle$ can be
neglected with respect to the sum of the other terms, which have the same
(positive) sign for $n$ large. The equality (\ref{eq:fredh}) is thus seen to
lead to a contradiction.  \hfill $\diamondsuit$\medbreak


When the function
$S^{n-1} \subset \ker(L-\lambda^* I)
  \to \mathbb R:x \mapsto h_0(x,\alpha,\beta)$ is negative, except at
one point
$x=x^*$,  the result of Theorem
\ref{theo:5} is fairly classical. The point $(\alpha, \beta)$ then lies on
the set (in general, a curve) denoted by $F^+$ in Section
\ref{section:prelim}, and
the condition on the degree is  automatically satisfied. For instance, when
$\dim \,\ker(L-\lambda^* I)=1$, the conclusions can be seen as a
particular case of results of Gallou\"et and
Kavian \cite{gak}. However, Theorem
\ref{theo:5} can also be used when $\lambda^*$ is an eigenvalue of
multiplicity greater than $1$, and when the
point $(\alpha,\beta)$ belongs to a Fu\v c\'\i k curve that lies between the
sets $F^-$ and $F^+$.
Such types of spectra have been considered in \cite{bfs}.
The application of Theorem \ref{theo:5} to that situation is illustrated
by the following example.

\paragraph{Example 1.} Fu\v c\'\i k curves for the following boundary
value problem have been studied in \cite{bfs}.
\begin{eqnarray}
&u^{(4)} +(m^2+n^2) u'' = \alpha u^+- \beta u^-,&
\label{eq:ode4}\\
&u(0)=u(\pi)=0, u''(0)=u''(\pi)=0 \,.&\label{eq:bc}
\end{eqnarray}
We choose here $m=6$ and $n=13$. The value $\lambda^* =m^2n^2=6084$ is an
eigenvalue of multiplicity $2$ for the operator
$$ L:\mathop{\rm dom} L \subset L^2((0,\pi);\mathbb R) \to 
L^2((0,\pi);\mathbb R):u \mapsto u^{(4)}
+(m^2+n^2)u''\,.$$
We take $\mathop{\rm dom} L= H^4((0,\pi);\mathbb R)$, the real 
Sobolev space of order 4.
The eigenspace associated with $\lambda^*$ is
spanned by the functions  $\sin mx \sin nx$. It has been shown in
\cite{bfs} that
four Fu\v c\'\i k curves pass through the point $(\lambda^*,
\lambda^*);$ their respective slopes at that point are $-0.7232,
-6/7,-7/6,-1.3828$.
We will assume that
$(\alpha,\beta)$ lies on the curve of slope $-7/6$. We expect hypothesis
(H5) of Theorem \ref{theo:5} to hold, although the
verification appears difficult.  On the other hand, the condition on the degree
is seen to be verified (at least for
$|\beta -\alpha |$ small). Indeed, the degree, with respect to open bounded
sets
containing
$0$, of  $x \mapsto c_0(x, \alpha+\varepsilon  , \beta+\varepsilon)$ is
equal to
$-1$, for
$\varepsilon $ small (see \cite{bfs}). Assuming that $f$ satisfies
the condition in Theorem \ref{theo:5} (only one
function $\varphi$ needs to be considered here),
we conclude that equation (\ref{eq:1}) has at least one solution, provided
the condition (H5) is indeed satisfied.


\section{Fu\v c\'\i k spectrum reduced to a curve} \label{section:H3}

The Fu\v c\'\i k spectrum turns out to be particularly simple when
(H3) holds. In this case, it results from Lemma \ref{lemma:x3}
that the sets $F^-$ and $F^+$ defined in Section \ref{section:prelim}
coincide and no other point of the
Fu\v c\'\i k spectrum is  contained in $I \times I$.
The condition (H3) appears in \cite{bfs} and, under a different form, in
\cite{fab}. It is shown there that the condition is satisfied for periodic
boundary
value problems for ordinary differential equations, when the operator $L$ is
autonomous (see \cite{bfs} or \cite{fab} for a precise statement).


Let $f(t,u)$ satisfy the Carath\'eodory conditions; i.e., $f(\cdot,u)$ is
measurable for all $u \in \mathbb R$, $f(t,\cdot)$ is continuous for a.e.
$t\in\Omega$. Moreover, assume that there exists
$K \in  L^2 (\Omega; \mathbb{ R})$,
such that
$$ |f(t,u)| \leq K(t), \mbox{ for all } u \in \mathbb R\,.$$
Let
$$g_{\pm}(t) = \liminf_{u \to \pm \infty} f(t,u), G_{\pm}(t) = \limsup_{u
\to \pm \infty}f(t,u).$$
Assume that $G^+-g_- \in L^{\infty} (\Omega; \mathbb{ R })$.
For $\varepsilon \geq 0, $ and with an arbitrary choice of $f_+,f_-$ satisfying
\begin{equation}
g_-(t) \leq f_-(t) \leq G_-(t),\, g_+(t) \leq f_+(t) \leq G_+(t) ,
\label{eq:set}
\end{equation}
we define $f_{\varepsilon}$ as in Section 2 by
\begin{equation}
f_{\varepsilon} (t,u)= \frac{1}{2}\, \left[f_+(t) +f_-(t)\right] +
\frac{1}{\pi}\, \left[f_+(t) -f_-(t)\right] \arctan(\varepsilon
u)\,.\label{eq:fe}
\end{equation}
To $f$, we associate the mapping
\begin{equation} N : L^2 (\Omega; \mathbb{ R }) \rightarrow L^2
(\Omega; \mathbb{ R }):u \mapsto \alpha u^+ -\beta u^- +f(\cdot,u)\;;
\label{eq:nn2}
\end{equation}
$N_{\varepsilon}$ is defined as before by
\begin{equation}
N_{\varepsilon} : L^2 (\Omega; \mathbb{ R }) \rightarrow L^2
(\Omega; \mathbb{ R }):u \mapsto \alpha u^+ -\beta u^-
+f_{\varepsilon}(\cdot,u).
\label{eq:nne2b}
\end{equation}

Under (H3), existence results will be obtained, based on the
computation of the coincidence degree
$D((L,N), B_r)$, with respect to balls $B_r \subset L^2 (\Omega; \mathbb{ R
})$, centered at $0$ and of
large radius
$r$ (see \cite{maw} for a definition of the coincidence degree).
Under conditions to be given below, the computation of that degree will be
shown to reduce to the
computation of the Brouwer's degree of the mapping
\begin{equation} c_{f_+,f_-}:\ker(L-\lambda^* I) \to \ker(L-\lambda^* I):
x \mapsto
\nabla_x\left[
\langle f_+,\varphi_x^+ \rangle -\langle
f_-,\varphi_x^- \rangle \right],\label{eq:defh}
\end{equation}
with respect to the unit ball in $\ker(L-\lambda^* I)$ (since the function
$c_{f_+,f_-}$ is not
likely to be continuous at $0$, the degree is to be understood as the
degree of a continuous extension to
the unit ball, of a restriction of $c_{f_+,f_-}$ to the unit sphere).
Notice that the mapping
$c_{f_+,f_-}$ is homogeneous of degree $0$, so that the Brouwer's degree
of $c_{f_+,f_-}$ is
the same
with respect to all balls centered at $0$. The precise statement of the
relation
between the two degrees is given below in Theorem \ref{theo:degree}. It can
be seen as an extension,
for the
resonance with respect to the Fu\v c\'\i k spectrum, of a result of
Krasnosel'skii
\cite{kra} for the ``classical'' resonance (the case $\alpha=\beta)$.

When (H3) is satisfied, it results from Lemma
\ref{lemma:1} that, with
$(\alpha,\beta) \in \Sigma(L)
\cap (I \times I)$, there exists a \emph{ unique} solution $\varphi_x$ of
(\ref{eq:2}) such that
$P\varphi_x = x$.
We will use a weakened version of assumption (H4):
\begin{description}

\item{(H4')} For some $p>2$, $\varphi_x \in L^p(\Omega;\mathbb R)$ for every
$x \in \ker (L-\lambda^* I)$ and the mapping
$\ker (L-\lambda^* I) \to L^p(\Omega;\mathbb R) : x \mapsto \varphi_x$
is Lipschitz continuous.
\end{description}
This will allow to consider non-elliptic operators as the wave
operator (see Example 2).

\begin{theorem} \label{theo:degree}
Assume that hypotheses (H1), (H2), (H3), (h4'), and (H5) hold. Also
assume that $L$ has a compact
resolvent and that $f(t,u)$ satisfies the hypotheses given above.
Let $N$ and $c_{f_+,f_-} $ be defined by (\ref{eq:nn2}) and  (\ref{eq:defh}),
respectively. Assuming $c_{f_+,f_-}(x) \neq 0$ for any $f_+, f_-$
satisfying (\ref{eq:set}) and any $x  \neq 0, $ the
coincidence degree
$D((L,N), B_r)$,
with respect to sufficiently large balls $B_r$ is equal, up to the sign, to the
Brouwer's degree $$d_B(c_{f_+,f_-},B_1 \cap \ker(L-\lambda^* I),0).$$
\end{theorem}

Of course, the proof needs to show that the above degree is independent of
$f_+, f_-$ satisfying (\ref{eq:set}).

Some preliminary steps are needed for the proof of Theorem
\ref{theo:degree}. We start with a lemma
stating that the mapping $x \to \varphi_x$ has a strong Fr\'echet
derivative, denoted $\varphi'_x:
\ker(L-\lambda^* I) \to L^2 (\Omega; \mathbb{ R })$, i.e.
$$ \|\varphi_y - \varphi_z - \varphi'_x(y-z) \| = o(\|y-z\|) \mbox{ for } y
\to x, z \to x.$$

\begin{lemma} \label{lemma:nx3} Let $L,\alpha,\beta$ satisfy hypotheses
$(H_1 ),(H_2 ),(H_3), (H_4'), (H_5)$.
Then, the function $x \mapsto \varphi_{x} $ admits
a strong Fr\'echet derivative $\varphi'_{x}: \ker(L-\lambda^* I) \to L^2
(\Omega;
\mathbb{ R })$ at $x \neq 0$,
$w_h = \varphi'_{x }h$ being the unique solution of
\begin{eqnarray}
&L w_h = (I-P) [\alpha \chi (\varphi_{x}^+)  +\beta
\chi  (\varphi_{x}^- )]w_h +\lambda^*h ,&\label{eq:der4}\\
&P w_h= h\label{eq:der4b}\,.&
\end{eqnarray}
Moreover, $w_h$ is also a solution of
\begin{equation}
L w_h = [\alpha \chi (\varphi_{x}^+)  +\beta
\chi  (\varphi_{x}^- )]w_h \label{eq:der4c}
\end{equation}
and, for fixed $h$, the function $x \mapsto \varphi'_{x} h$ is
continuous with respect to $x$.
\end{lemma}

\paragraph{Proof.} Notice first that, by the same arguments as for Lemma
\ref{lemma:1}, the system
(\ref{eq:der4}), (\ref{eq:der4b}) defines a unique function $w_h;$
moreover, it is clear that
$w_h$ is a bounded linear function of  $h \in \ker(L-\lambda^* I)$. Let us
prove that $w_h$ is indeed
the strong Fr\'echet derivative of $x \mapsto \varphi_x$. By definition, we
have
\begin{eqnarray*}
L \varphi_x  &=& \alpha \varphi_{x}^+  -\beta \varphi_{x}^-\;, \;P
\varphi_{x}=x,\\
L \varphi_y  &=& \alpha \varphi_{y}^+  -\beta \varphi_{y}^-\;, \;P
\varphi_{y}=y.
\end{eqnarray*}
With $h=y-z$, and $w_h=w_{y-z}$ defined by (\ref{eq:der4}),
(\ref{eq:der4b}), we get, by subtraction,
\begin{eqnarray}
  L (\varphi_y- \varphi_z - w_{y-z})
&=& \alpha (I-P) [\varphi_{y}^+ -\varphi_{z}^+
-\chi(\varphi_{x}^+) (\varphi_{y}-\varphi_{z})] \nonumber \\
&& + \alpha (I-P) \chi(\varphi_{x}^+)
(\varphi_{y}-\varphi_{z}-w_{y-z})  \label{eq:lo} \\
& &-\beta (I-P) [\varphi_{y}^- - \varphi_{z}^-
-\chi(\varphi_{x}^-) (\varphi_{y}-\varphi_{z})] \nonumber \\
&&-\beta (I-P) \chi(\varphi_{x}^-) (\varphi_{y}-\varphi_{z}-w_{y-z}),
\nonumber
\end{eqnarray}
taking into account that $P(\alpha \varphi_{y}^+  -\beta
\varphi_{y}^-)=\lambda^* y, $
$ P(\alpha \varphi_{z}^+  -\beta \varphi_{z}^-)=\lambda^* z$. On the other
hand, we have
\begin{equation}
P( \varphi_{y} -\varphi_{z} -w_{y-z})=0.
\label{eq:p1}
\end{equation}
By  hypotheses (H4') and (H5), the mappings $u \mapsto u^+$ and $u
\mapsto u^-$ have
strong Fr\'echet derivatives at $\varphi_x$, as functions from
$L^p(\Omega; \mathbb R)$ to $
L^2(\Omega; \mathbb R)$ (see \cite{bfs}), their  respective derivatives
being the mappings $h \mapsto
\chi(\varphi_x^+) h $ and $h \mapsto
-\chi(\varphi_x^-) h $.  This means that
$$
\frac{ \|u^+ -v^+ - \chi(\varphi_x^+)(u-v)\|}{\|u-v\|_{L^p}} \; \to 0
\mbox{ for } \; u
\stackrel{L^p}{\to} \varphi_x, v \stackrel{L^p}{\to} \varphi_x\;.$$
  But, the mapping $\ker(L-\lambda^* I) \to L^p(\Omega; \mathbb
R):x \mapsto \varphi_x$ is Lipschitz continuous; consequently,
\begin{equation}
\frac{ \|\varphi_y^+ -\varphi_z^+ -
\chi(\varphi_x^+)(\varphi_y-\varphi_z)\|}{\|y-z\|} \;
\to 0
\mbox{ for } \; y \to x, z \to x\,. \label{eq:fp}
\end{equation}
A similar result holds for the negative parts. On the other
hand, it follows from
(\ref{eq:lo}), (\ref{eq:p1}), using arguments similar to those of Lemma
\ref{lemma:1}, that, for some
$C_1 >0$,
\begin{eqnarray*}
\|\varphi_y- \varphi_z - w_{y-z}\| &\leq &C_1 \left[ \| \varphi_{y}^+ -
\varphi_{z}^+
-\chi(\varphi_{x}^+) (\varphi_{y}-\varphi_{z})\| \right.\\
  & & \left.+ \|\varphi_{y}^- - \varphi_{z}^-
-\chi(\varphi_{x}^-) (\varphi_{y}-\varphi_{z})\|\right]
\end{eqnarray*}
which, combined with (\ref{eq:fp}), shows that the mapping $ h \mapsto w_h$
is the strong Fr\'echet
derivative of $x \mapsto \varphi_x$.

On the other hand, $\varphi_x$ being a solution of (\ref{eq:2}), we have
\begin{equation}
P[\alpha \varphi_x^+ - \beta \varphi_x^-] = \lambda^* x, \mbox{ for all } x \in
\ker(L-\lambda^* I).\label{eq:id}
\end{equation}
Since $\chi(\varphi_x^+), - \chi(\varphi_x^-)$ are the Fr\'echet
derivatives, at $\varphi_x$,
of $u
\mapsto
u^+$ and $u \mapsto u^-$ respectively, we see, by
differentiating (\ref{eq:id}), that
$$ P[\alpha \chi(\varphi_x^+) \varphi'_x h + \beta \chi(\varphi_x^-)
\varphi'_x h] = \lambda^* h,$$
which derives (\ref{eq:der4c}) from (\ref{eq:der4b}).

It remains to show that $\varphi'_x h $ is continuous with respect to
$x$. From (\ref{eq:der4}), (\ref{eq:der4b}), we have
\begin{eqnarray*}
L (\varphi'_x  h- \varphi'_y h) &=& (I-P) [\alpha \chi(\varphi_{x}^+)  +\beta
\chi(\varphi_{x}^-)(\varphi'_x h - \varphi'_y h)]\\
&& + \alpha(I-P) [\chi(\varphi_{x}^+) - \chi(\varphi_{y}^+)] \varphi'_y  h\\
& &+ \beta (I-P) [\chi(\varphi_{x}^-) - \chi(\varphi_{y}^-)] \varphi'_y  h,\\
P(\varphi'_x  h- \varphi'_y h)&=&0.
\end{eqnarray*}
Using again arguments like in Lemma \ref{lemma:1}, we obtain, for some
constant $C_2$,
$$
\|\varphi'_x  h- \varphi'_y h\| \leq C_2 [\|\chi(\varphi_{x}^+) -
\chi(\varphi_{y}^+)\| +
\|\chi(\varphi_{x}^-) - \chi(\varphi_{y}^-)\|]\,.$$ The conclusion then
follows from hypothesis
(H5). \hfill $\diamondsuit$


\begin{lemma} \label{lemma:nx4} Let $L,\alpha,\beta$ satisfy hypotheses
$(H_1 ),(H_2 ),(H_3), (H_4')$ and (H5).  Let $\{x_n\} \subset
\ker(L-\lambda^* I)$
be such
that $\|x_n\| \to \infty, x_n/\|x_n\| \to x^*$. Then, with
$f_{\varepsilon}$ defined
by (\ref{eq:fe}) (assuming $\varepsilon $ fixed, but small enough),
\begin{equation}
\lim_{n \to \infty} c(x_n,f_{\varepsilon},\alpha,\beta)= - c_{f_+,f_-}(x^*).
\label{eq:nx4}
\end{equation}
\end{lemma}

\paragraph{Proof.} With the notations of Section \ref{section:prelim},
we have
$$ Lu_x = \alpha u^+_x - \beta u^-_x +  f_{\varepsilon}(\cdot,u_x) + c
(x,f_{\varepsilon},\alpha,\beta) ,$$
and, by the previous lemma,
$L(\varphi'_{x }h) = [\alpha \chi (\varphi_{x}^+)   +\beta
\chi  (\varphi_{x}^- )] (\varphi'_{x}h)$.
It then follows from the self-adjointness of $L$, that
\begin{eqnarray}
  0 &=& \langle f_{\varepsilon}(\cdot,u_x), \varphi'_{x}h \rangle +
\langle c (x,f_{\varepsilon},\alpha,\beta), \varphi'_{x}h \rangle \nonumber \\
& & +(\alpha - \beta) \langle u_x^+ \chi(\varphi_x^-),\varphi'_{x}h
\rangle +(\beta -\alpha) \langle u_x^- \chi(\varphi_x^+) ,\varphi'_{x}h
\rangle.\label{eq:ps2}
\end{eqnarray}
To estimate the last two terms in the above formula, we will use
inequalities like
\begin{eqnarray}
\langle u_x^+ \chi(\varphi_x^-) ,\varphi'_{x}h
\rangle & \leq & \big( \int_{u_x >0, \varphi_x <0} u_x^2 \big)^{1/2}
\big( \int_{u_x >0, \varphi_x <0} (\varphi'_{x}h)^2 \big)^{1/2}
\nonumber \\
& \leq & \|u_x - \varphi_x \| \big( \int_{u_x >0, \varphi_x <0}
(\varphi'_{x}h)^2 \big)^{1/2}.
\label{eq:est}
\end{eqnarray}
It is easy to show that $\|u_x - \varphi_x\|$ is bounded, independently of
$x$. Hence, we will have
\begin{equation}
\lim_{n \to \infty} \langle u_{x_n}^+ \chi(\varphi_{x_n}^-) ,\varphi'_{{x_n}}h
\rangle = 0, \label{eq:la1}
\end{equation}
if we can show that
\begin{equation}
\lim_{n \to \infty}  \int_{u_{x_n} >0, \varphi_{x_n} <0}
(\varphi'_{{x_n}}h)^2  =
0. \label{eq:ob}
\end{equation}
With $\|x_n\| \to \infty, x_n/\|x_n\|
\to x^*$, we have $(u_{x_n} - \varphi_{x_n})/\|x_n\| \to 0$,
$\varphi_{x_n}/\|x_n\| \to \varphi_{x^*}$,  and using
(H5), it follows that
$$\lim_{n\to\infty}\mathop{\rm meas}\{t\in
\Omega \mid u_{x_n}(t) \varphi_{x_n}(t) <0\} =0,
$$
which implies
$$ \lim_{n \to \infty}  \int_{u_{x_n} >0, \varphi_{x_n} <0}
(\varphi'_{x^*}h)^2  =0\,.
$$
The limit (\ref{eq:ob}) then follows from the continuity of
$x \mapsto\varphi_x'$.
Using (\ref{eq:la1}) and
a similar result for $\langle u_{x_n}^- \chi(\varphi_{x_n}^+)
,\varphi'_{{x_n}}h
\rangle $, we deduce from (\ref{eq:ps2}) that
\begin{equation}
\lim_{n \to \infty} \left[ \langle f_{\varepsilon}(\cdot,u_{x_n}),
\varphi'_{{x_n}}h
\rangle +   \langle c ({x_n},f_{\varepsilon},\alpha,\beta),
\varphi'_{{x_n}}h \rangle
\right]=0.\label{eq:42b}
\end{equation}
But, $f_{\varepsilon}(t,u_{x_n}) $, converges pointwise to $
f_+(t)
\chi(\varphi_{x^*}^+) + f_-(t) \chi(\varphi_{x^*}^-)$, and, by Lebesgue
dominated
convergence theorem, convergence is also true in the $L^2$-norm. On the
other hand,
\begin{eqnarray*}  \langle c ({x},f_{\varepsilon},\alpha,\beta),
\varphi'_{{x}}h
\rangle &=&
\langle c ({x},f_{\varepsilon},\alpha,\beta), P(\varphi'_{{x}}h )\rangle \\
&=& \langle c ({x},f_{\varepsilon},\alpha,\beta), h )\rangle.
\end{eqnarray*}
Putting together those observations, we conclude that
\begin{eqnarray*}
\lim_{n \to \infty}   \langle
c({x_n},f_{\varepsilon},\alpha,\beta),
h \rangle
  &=&  - \langle f_+(t)
\chi(\varphi_{x^*}^+) + f_-(t) \chi(\varphi_{x^*}^-), \varphi'_{{x^*}}h \rangle
\end{eqnarray*}
from which (\ref{eq:nx4}) follows. \hfill $\diamondsuit$ \smallskip

We are now ready to prove Theorem \ref{theo:degree}.


\paragraph{Proof of Theorem \ref{theo:degree}.}
Let us choose
arbitrarily $f_+,f_-$ satisfying
(\ref{eq:set}); with that choice of $f_+,f_-$, let
$f_{\varepsilon},N_{\varepsilon}$ be defined as
above by (\ref{eq:fe}), (\ref{eq:nne2b}) respectively. We will first show
that, for
$r$ sufficiently large, the following equality holds
\begin{equation}
D((L,N),B_r) = D((L,N_{\varepsilon}),B_r), \label{eq:cod}
\end{equation}
between the coincidence degree involving the mappings $N$ and
$N_{\varepsilon}$. By the
invariance property of the degree with respect to homotopies, that result is
established if we can find an a priori bound for the solutions of
\begin{equation}
Lu = s Nu +(1-s) N_{\varepsilon} u, \quad s \in [0,1].
\label{eq:b11}
\end{equation}
By contradiction, assume that there exist sequences $\{u_n\},\{s_n\}$, with
$\|u_n\|
\to \infty , s_n \in [0,1]$, such that
$$ Lu_n = s_n Nu_n +(1-s_n) N_{\varepsilon} u_n.$$
Let $x_n=P u_n;$ extracting if necessary a subsequence, we can assume that
$x_n/\|x_n\|$ converges to some $x^* \in \ker(L-\lambda^* I) $ and $s_n$
converges to some $s^* \in
[0,1]$. It is easy to prove
that $\|x_n\| \to \infty $ and that $u_n/\|x_n\| \to \varphi_{x^*}$, using
the fact that $f$ and
$f_{\varepsilon}$
are bounded by an $L^2$-function. Since, by Lemma \ref{lemma:1},
\begin{equation}
Lu_n = N_{\varepsilon} u_n +c(x_n, f_{\varepsilon}, \alpha,\beta),
\label{eq:b12}
\end{equation}
we get, subtracting  (\ref{eq:b12}) from (\ref{eq:b11}),
$$ 0= s_n(N u_n - N_{\varepsilon} u_n) - c(x_n, f_{\varepsilon},
\alpha,\beta),$$
i.e.
$$ 0= s_n[f(\cdot, u_n) - f_{\varepsilon}(\cdot, u_n) ] - c(x_n,
f_{\varepsilon},
\alpha,\beta).$$
Multiplying the last equality by $\varphi'_{x_n} h$, for an arbitrary $h
\in \ker(L-\lambda^* I)$,  we
get, using (\ref{eq:42b}) and Lemma 4,
\begin{equation}
  s^* \lim_{n \to \infty} \langle f(\cdot,u_n), \varphi'_{x_n} h \rangle +
(1-s^*)
\langle f_+ \chi(\varphi_{x^*}^+) + f_- \chi(\varphi_{x^*}^-) ,
\varphi'_{x^*} h
\rangle=0\,.\label{eq:sa4}
\end{equation}
On the other hand, passing if necessary to a further subsequence, we can
assume that $f(\cdot,u_n)$
converges weakly to some function
$$ \tilde{f}_+ \chi(\varphi_{x^*}^+) + \tilde{f}_- \chi(\varphi_{x^*}^-),$$
with
\begin{eqnarray*}
  \liminf_{u \to - \infty} f(t,u) \leq &\tilde{f}_-(t)& \leq \limsup_{u \to
- \infty} f(t,u),\\
\liminf_{u \to + \infty} f(t,u) \leq &\tilde{f}_+(t)& \leq \limsup_{u \to +
\infty} f(t,u).
\end{eqnarray*}
It then follows from (\ref{eq:sa4}) that
\begin{equation}
s^* \langle \tilde{f}_+ \chi(\varphi_{x^*}^+) + \tilde{f}_-
\chi(\varphi_{x^*}^-), \varphi'_{x^*} h
\rangle + (1-s^*)
\langle f_+ \chi(\varphi_{x^*}^+) + f_- \chi(\varphi_{x^*}^-), \varphi'_{x^*} h
\rangle=0\,.\label{eq:sa5}
\end{equation}
Let
$$ f^*= (s^*\tilde{f}_+ + (1-s^*) f_+) \chi(\varphi_{x^*}^+) +
(s^*\tilde{f}_- +(1-s^*) f_-)
\chi(\varphi_{x^*}^-) \,;$$
The equation (\ref{eq:sa5}) can also be written
$$ \nabla_x   \left[
\langle (f^*_+,\varphi_{x^*}^+ \rangle -\langle
(f^*_-,\varphi_{x^*}^- \rangle \right] \rangle =0$$
or
$$c_{f^*_+,f^*_-}(x^*)=0.$$
  But, this contradicts the hypotheses and proves (\ref{eq:cod}).

It remains to show
that, for $r$
large,
$$D((L,N_{\varepsilon}),B_r) = \pm d_B(c_{f_+,f_-}, B_1 \cap
\ker(L-\lambda^* I),0).$$
Denoting by $K:\mbox{Range}(L-\lambda^* I) \to \mbox{Range}(L-\lambda^* I)$ the
right inverse of
$L-\lambda^* I$ with respect to $P$, and using a Lyapunov-Schmidt
decomposition, the
problem
$Lu = N_{\varepsilon} u$ is transformed into
$$u=Pu +K(I-P) N_{\varepsilon}   +
c(x_n, f_{\varepsilon},
\alpha,\beta).$$ By the homotopy
$$u=Pu +s K(I-P) N_{\varepsilon}  +
c(x_n, f_{\varepsilon},
\alpha,\beta),$$
it is seen that the Leray-Schauder degree, with respect to large balls of
$$u \mapsto u-Pu -K(I-P) N_{\varepsilon}  -
c(x_n, f_{\varepsilon},
\alpha,\beta)$$
is the same as that of
$$u \mapsto u-Pu   - c(x_n, f_{\varepsilon},
\alpha,\beta).$$ By a product formula, this last degree is the same, for
$r$ large,
as the Brouwer degree $d_B(c(\cdot, f_{\varepsilon},
\alpha,\beta),B_r \cap \ker(L-\lambda^* I),0)$. The conclusion then follows
from (\ref{eq:nx4}). \hfill $\diamondsuit$\medbreak

Working as in \cite{bfs} (see Theorem 10), the degree $d_B(c_{f_+,f_-},B_1 \cap
\ker(L-\lambda^* I),0)$ can be computed from the indices of critical
points, with
negative critical values, of the function
$$ \Psi : x \mapsto \langle f_+, \varphi_x^+ \rangle - \langle f_-, \varphi_x^-
\rangle,$$ restricted to the unit sphere $S^{n-1}$ in $\ker(L-\lambda^*
I)$. More
precisely, the following corollary holds.

\begin{cor}\label{cor:cor1}
Under the hypotheses of Theorem \ref{theo:degree}, the coincidence degree \\
$D((L,N),B_r)$, with respect to large balls $B_r$, is, up to the sign,
given by
\begin{equation}
  1- \sum\mathop{\rm ind}_x \, \nabla_x \Psi_{|S_{n-1}}(x)\,,\label{eq:ind}
\end{equation}
where the summation is taken over $x \in S^{n-1}$ such that
$\nabla_x({\Psi}_{|S^{n-1}})(x)=0$ and $\Psi_{|S_{n-1}}(x)<0$.
As a consequence, if the sum of the indices in (\ref{eq:ind}) is different
from $1$,
problem (\ref{eq:1b}) has at least one solution.
\end{cor}

When $\dim \ker(L - \lambda^* I) = 1$, letting $\ker(L - \lambda^* I) =
\mathbb R x^*$, the
result must be interpreted as follows: if $\Psi(x^*) \Psi(-x^*) >0, $ then
the degree is equal to $\pm
1;$ whereas, when $\Psi(x^*) \Psi(-x^*) <0, $ the degree is equal to $0$.

The following result is an immediate consequence of the above corollary.
\begin{cor}
Under the hypotheses of Theorem \ref{theo:degree}, assume that, with
$g_+,G_-$ defined by
(\ref{eq:set}),
\begin{equation} \langle g_+, \varphi_x^+ \rangle - \langle G_-, \varphi_x^-
\rangle >0, \mbox{ for any } x \in
\ker(L-\lambda^* I), x
\neq 0.
\label{eq:lala}
\end{equation}
Then, problem (\ref{eq:1b}) has at least one solution.
\end{cor}

The condition (\ref{eq:lala}), in which the sign could be reversed (with
$g_+, G_-$ replaced
respectively by $G_+,g_-)$, is a condition
of Landesman-Lazer type. It has been obtained by Gallou\"et and Kavian
\cite{gak} for
the case $\mathop{\rm dim}\ker(L-\lambda^* I)=1$. When $f$ does not depend on
$u$, it amounts to the condition
appearing in Theorem \ref{theo:5}, i.e. $\langle f, \varphi \rangle >0$,
for any nontrivial solution
$\varphi$ of (\ref{eq:2}).

Actually, when $\mathop{\rm dim}\ker(L-\lambda^* I)=1$, existence 
results can be
obtained by the above
approach only when $\Psi$  always takes the same sign (for any $f_+, f_-$
satisfying
(\ref{eq:set}) and any $x \in \ker(L-\lambda^* I), x \neq 0), $ a situation
which corresponds to
Landesman-Lazer conditions. This is no longer true for higher dimensional
kernels. It can be seen
in particular when
$\mathop{\rm dim}\ker(L-\lambda^* I)=2$, in which case
  the result of Corollary \ref{cor:cor1} takes a very simple form. To state
it, we let
$\{ v^{(1)}, v^{(2)} \}$ be an orthonormal basis of
$\ker (L - \lambda^{\ast} I)$,  define $z_{\theta}$ by
$$z_{\theta} = \cos \theta \,v^{(1)} + \sin \theta \,v^{(2)}\,,$$
and use the abbreviation $\varphi_{\theta}$ for $ \varphi_{z_{\theta}}$.


\begin{cor} \label{theo:5b}
Let the hypotheses of Theorem \ref{theo:degree} be satisfied, with
$\mathrm{dim}\ker(L-\lambda^* I)=2. $  Assume moreover that, for any
$f_+,f_-$ satisfying
(\ref{eq:set}), the  function
$\Psi :
\theta \mapsto \langle f_+,\varphi^+ _{\theta} \rangle - \langle
f_-,\varphi^-_{\theta}
\rangle$ has
only  simple zeros, the number of zeros in $[0,2 \pi)$ being different from
$2$.
Then,  equation (\ref{eq:1b}) has at least one solution.
\end{cor}


For equation (\ref{eq:1}), the arguments can be simplified with respect to the
treatment above. Indeed, the problem can then be immediately reduced to a
computation of degree in $\ker(L-\lambda^* I)$, by a Lyapunov-Schmidt
decomposition.
In fact, (\ref{eq:1}) has a solution if and only if there exists $x \in
\ker(L-\lambda^* I)$ such that $c(x,f,\alpha,\beta)=0$ (for equation
(\ref{eq:1b}),
the difference is that this reduction is not possible in general). It is not
necessary to compute a coincidence degree for the pair $(L,N)$, and, hence, the
compactness hypothesis can be removed, leading to the following result.


\begin{theorem} \label{theo:ex}
Assume that hypotheses (H1), (H2), (H3), (H4'), and (H5) hold, and
that
$f$ is a given element of $L^2 (\Omega; \mathbb{ R })$.
For $x \in
\ker(L-\lambda^* I)$, let $\Psi(x) = \langle f, \varphi_x \rangle$.
Then, if
$$\sum \mathop{\rm ind}_x \, \nabla_x \Psi_{|S_{n-1}}(x) \neq 1,$$
where the summation is taken over $x \in S^{n-1}$ such that
$\nabla_x({\Psi}_{|S^{n-1}})(x)=0$ and $\Psi_{|S_{n-1}}(x)<0$,
then equation (\ref{eq:1}) has at least one solution.
\end{theorem}

Again, in the case $\mathop{\rm dim}\ker(L-\lambda^* I)=2, $  a 
particularly simple
existence
condition can be written.

\begin{cor}  \label{theo:ex2}
Let the hypotheses of Theorem \ref{theo:ex} be satisfied, with
$\mathrm{dim}\ker(L-\lambda^* I)=2. $  Assume moreover that the  function
$\Psi :
\theta \mapsto \langle f,\varphi_{\theta} \rangle$ has only simple zeros,
the number
of zeros in
$[0,2
\pi)$ being different from $2$. Then,  equation (\ref{eq:1}) has at least one
solution.
\end{cor}

The results of \cite{fa-fo}, concerning  the $2 \pi-$periodic boundary-value
problem for the equation
\begin{equation}
u'' + \alpha u^+ -\beta u^-= f(t),\label{eq:fafo}
\end{equation}
where $f$ is $2 \pi-$periodic, can be seen as an application of the 
above corollary (at
least for the points $(\alpha,\beta) $ in the Fu\v c\'\i k spectrum 
``not too far'' from the diagonal in
the $(\alpha,\beta)$-plane). Assuming that
$$ \frac{1}{\sqrt \alpha} + \frac{1}{\sqrt \beta} = \frac{2}{n},$$
for some integer $n,$
the function $\varphi_{\theta}$ can be  defined by $\varphi_{\theta}(t) =
\varphi(t+\theta)$,
$\varphi$ being a solution of $u'' + \alpha u^+ -\beta u^-=0$. We can choose
$$
\varphi(t)=
\left\{
\begin{array}{ll}
\frac{1}{\sqrt{\alpha}}\sin (\sqrt{\alpha} t) & (t\in
[0,\frac{\pi}{\sqrt{\alpha}}]),\\[5pt]
-\frac{1}{\sqrt{\beta}}\sin (\sqrt{\beta}(t-\frac{\pi}{\sqrt{\alpha}})) & (t\in
[\frac{\pi}{\sqrt{\alpha}}, \frac{2\pi}{n}]).
\end{array}
\right.
$$
Computing
$$ \Psi(\theta) = \int_0^{2\pi} f(t) \varphi(t+\theta) \, dt,$$
we can assert that (\ref{eq:fafo}) has a solution, provided that $\Psi $
has only simple zeros,
the number
of zeros in
$[0,2 \pi)$ being different from $2$.

\medbreak

Another application of Corollary \ref{theo:ex2} is given below.

\paragraph{Example 2.} The above result can be applied to the problem
\begin{eqnarray}
&u_{tt}-u_{xx}= \alpha u^+ -\beta u^- +f\,, \label{eq:wave} \\
&u(0,t) =u(\pi,t)=0\,, \mbox{ for all } t \in [0,2\pi], \label{eq:pd1} \\
&u(x,t)=u(x,t+2\pi), \mbox{ for all }x \in [0,\pi],  t \in \mathbb
R\,,\label{eq:pd2}
\end{eqnarray}
where the function $f: [0,\pi] \times \mathbb R  \to \mathbb R: (x,t) \mapsto
f(x,t) $ is $2\pi$-periodic in $t$, and belongs to $L^2(\Omega; \mathbb R)$,
where $\Omega =(0,\pi) \times (0,2\pi)$. We define
$$ \varphi_{lm}(x,t)=
\left\{
\begin{array}{ll}
\frac{\sqrt{2}}{\pi} \, \sin(lx) \sin(mt), & l \in \mathbb N, m \in \mathbb
N\,,\\[5pt]
\frac{1}{\pi} \, \sin(lx),&l \in \mathbb N, m=0\,,\\[5pt]
\frac{\sqrt{2}}{\pi} \, \sin(lx) \cos(mt), & l \in \mathbb N, -m \in
\mathbb N\,,
\end{array}
\right.
$$
The set $\{\varphi_{lm}, l \in \mathbb N, m \in \mathbb Z\}$ forms an
orthonormal basis
in $L^2(\Omega; \mathbb R)$ and each $u \in L^2(\Omega; \mathbb R)$ admits
  a representation
$$
u=\sum_{l=1}^{\infty} \sum_{m=-\infty}^{\infty} u_{lm} \varphi_{lm},$$
with $u_{lm}=\langle u, \varphi_{lm} \rangle$. The abstract realization of
the wave
operator $u_{tt} - u_{xx}$, with the periodic-Dirichlet boundary conditions
(\ref{eq:pd1}), (\ref{eq:pd2}), is the linear operator
$L: \mathop{\rm dom} L\subset L^2(\Omega; \mathbb R)\to  L^2(\Omega; 
\mathbb R)$,
  defined by
$$ Lu= \sum_{l=1}^{\infty} \sum_{m=-\infty}^{\infty} (l^2-m^2) u_{lm}
\varphi_{lm},$$
where
$$\mathop{\rm dom} L =\left\{ u \in L^2(\Omega; \mathbb R)  \mid 
\sum_{l=1}^{\infty}
\sum_{m=-\infty}^{\infty} |l^2-m^2|^2\, |u_{lm}|^2 < \infty  \right\}.
$$
It is well known that $L$ is densely defined, self-adjoint, closed, that
$\mathop{\rm Im} L=(\ker L)^{\perp}$, and that $L$ has a pure point spectrum
$\sigma(L) $
made of eigenvalues:
$$\sigma(L) = \{ \lambda_{lm} =l^2-m^2 , l \in \mathbb N, m \in \mathbb Z\}.$$
It is not hard to see that each eigenvalue $\lambda_{lm} \neq 0$ has finite
multiplicity,
$0$ is of infinite multiplicity, and $\sigma(L)$ can be written in the
following
form:
$$ \sigma(L)=\mathbb  Z \setminus \{-1,-4, 4m+2, m \in \mathbb Z\}.$$
In order to apply Corollary \ref{theo:ex2} to problem (\ref{eq:wave}),
(\ref{eq:pd1}), (\ref{eq:pd2}), we take, for instance, $\lambda^* = -3$,
and first
observe that hypothesis
$(H_1) $ is satisfied and that $\lambda^* = -3$ is of multiplicity $2$. The
eigenvalues closest to
$-3$ are
$-5$ and $0$, so that
$(H_2) $ is satisfied with $I = [-5+\varepsilon, -\varepsilon]$,
$\varepsilon$ being
positive. The hypothesis $(H_3) $ follows from the fact that, if $u(x,t)$ is a
solution of
\begin{equation}
u_{tt}-u_{xx}= \alpha u^+ -\beta u^-, \label{pdh}
\end{equation}
with the periodic-Dirichlet boundary value conditions (\ref{eq:pd1}),
(\ref{eq:pd2}),
the same is true for $u(x,t+\theta)$, for any $\theta \in [0,2\pi]$ (see
\cite{bfs}
for a similar argument applied to ordinary differential equations).
Actually, it
is easy to check that the Fu\v c\'\i k spectrum for that problem contains
the curve
$$\frac{1}{\sqrt{1-\alpha}}+ \frac{1}{\sqrt{1-\beta}}=1,$$
passing trough the point $(-3,-3), $ corresponding to the eigenvalue chosen
here.
The corresponding solutions of
(\ref{pdh}), (\ref{eq:pd1}), (\ref{eq:pd2}) can be written explicitly under
the form
$$\varphi_{\theta}(x,t) = \sin x \;v(t+\theta/2), \; \theta \in [0,2\pi],$$
$v$ being a nontrivial $\pi$-periodic solution of
$$ v'' + (1-\alpha)v^+ - (1-\beta)v^- = 0.$$
Thus, assumptions (H4'), (H5) are also satisfied.

Applying Corollary \ref{theo:ex2}, we then obtain the following
existence result.

\begin{cor}
Let $\alpha, \beta $ be in $I = [-5+\varepsilon, -\varepsilon]. $ Assume
that the
function
$$\Psi : \theta \mapsto \int_0^{2\pi}\!\!\!\int_0^{\pi} \sin x \, f(x,t)
  \;v(t+\theta) \,dx \, dt $$
has only simple zeros, the number of zeros in $[0,2\pi)$  being denoted by
$2z$.
Then, if $z\neq 1$, problem (\ref{eq:wave}), (\ref{eq:pd1}),
(\ref{eq:pd2}) has at least one solution.
\end{cor}

In particular, if $f(x,t) = G(x) H(t)$, the number $2z$ is equal to the
number of
zeros, in $[0,2\pi)$,  of the function
$$ \theta \mapsto \int_0^{2\pi}H(t) v(t +\theta)\,dt\;,$$
provided that $\int_0^{\pi} \sin x\,G(x) \, dx \neq 0$.


\section{A non-existence result.} \label{section:non}

The non-existence result presented below complements the existence results
of the previous sections.  It is inspired from a result
written by Lazer and McKenna \cite{lm1} for the $2 \pi$-periodic boundary
value problem for the differential equation
$$u'' + \alpha u^+ - \beta u^- = \cos (nx),$$
with $n$ an integer and
$$\frac{1}{\sqrt{a}} + \frac{1}{\sqrt{\beta}} = \frac{2}{n},
(n - 1)^2 < \alpha, \beta < (n + 1)^2.$$

\begin{theorem} \label{theo:6}
  Let $L,\alpha,\beta$ satisfy hypotheses
(H1), (H2), (H3).  Then, if
\begin{eqnarray}
\| Pf \| > \frac{| \beta - \alpha |}
{2 d - | \beta - \alpha |} \| (I - P) f \|,
\label{eq:(x_1 )}
\end{eqnarray}
where $ d = \;\mathrm{ dist } \left((\alpha + \beta)/2, \sigma (L) \setminus
\{ \lambda^{\ast} \} \right) $, the equation
$$Lu = \alpha u^+ - \beta u^- + f$$
has no solution.
\end{theorem}

\paragraph{Proof.} By contradiction, assume that $u$ is a
solution.  By hypothesis (H3), it is possible to find
$\tilde u \mathop{\rm dom} L$ such that
$$ L \tilde u = \alpha \tilde u^+ - \beta \tilde u^-, P \tilde u =
Pu.$$
Since $u - \tilde u \in [\ker (L-\lambda^* I)]^{\perp} $, letting $\mu =
(\alpha + \beta)/2$, we can write
$$(L - \mu I) (u - \tilde u) = (I - P) [(\alpha - \mu)
(u^+ - \tilde u^+) - (\beta - \mu) (u^- - \tilde u^-) + f ].$$
But, the function $u \mapsto (\alpha - \mu) u^+ - (\beta - \mu) u^-$
is Lipschitzian with Lipschitz constant
$ | \beta - \alpha | /2$, so that,
$$\| \tilde L^{-1}_{\mu} \|^{-1}
\| u - \tilde u \| \leq \| (I - P) f \| +
\frac{| \beta - \alpha |}{2} \,
\| u - \tilde u \|,$$
with
$\tilde L_{\mu} = (L - \mu I)_{|[\ker (L - \lambda^{\ast} I)]^{\perp} }$.
Since $\| \tilde L^{-1}_{\mu} \|^{-1} =
\mathop{\rm dist} (\mu,\sigma (L_{|[\ker (L - \lambda^{\ast}
I)]^{\perp} })) = d$ and since $d > | \beta - \alpha |/2$, we
obtain the relation
\begin{eqnarray}
\| u - \tilde u \| \leq
\frac{\| (I - P) f \|}{d - | \beta - \alpha |/2 }.
\label{eq:(x_2 )}
\end{eqnarray}
On the other hand, projecting  the equations
$$Lu = \alpha u^+ - \beta u^+ + f
\quad\mbox{and}\quad
L \tilde u = \alpha \tilde u^+ - \beta \tilde u^+$$
on $\ker(L-\lambda^* I)$, we get
\begin{eqnarray*}
P [(\alpha - \lambda^*)u^+ - (\beta - \lambda^*)u^- ] &=& - Pf, \\
P [(\alpha - \lambda^*) \tilde u^+ - (\beta - \lambda^*) \tilde u^- ] &=& 0,
\end{eqnarray*}
or
\begin{eqnarray*}
P [(\alpha - \mu) u^+ - (\beta - \mu) u^-] &=& - Pf +
(\lambda^*-\mu)Pu,\\ P [(\alpha - \mu) \tilde u^+ - (\beta - \mu) \tilde
u^- ] &=&
(\lambda^*-\mu)P\tilde u,
\end{eqnarray*}
from which we deduce by subtraction, using again the
Lipschitz constant for
$u \mapsto (\alpha - \mu) u^+ - (\beta - \mu) u^-$,
$$\| Pf \| \leq \frac{| \beta - \alpha |}{2}
\| u - \tilde u \|.$$
Combining this with (\ref{eq:(x_2 )}) leads to
$$\| Pf \| \leq \frac{| \beta -\alpha |/2}
{d - | \beta - \alpha |/2 } \| (I - P) f \|,$$
in contradiction with hypothesis (\ref{eq:(x_1 )}). \hfill $\diamondsuit$
\medbreak

Notice that, under the hypothesis of Theorem 5, if
$\dim (\ker (L - \lambda^{\ast} I)) = 2$, the confrontation with
Theorem 4 implies that the function
$\theta \mapsto \langle f, \varphi_{\theta} \rangle$ necessarily has two zeros
in $[0,2 \pi )$, if its zeros are simple.

\medbreak

\paragraph{Example 2'.} Coming back to the periodic-Dirichlet boundary
value problem
\begin{eqnarray*}
&u_{tt}-u_{xx}= \alpha u^+ -\beta u^- +f\,,  \\
&u(0,t) =u(\pi,t)=0\,, \mbox{for all } t \in [0,2\pi],  \\
&u(x,t)=u(x,t+2\pi), \mbox{ for all }x \in [0,\pi],  t \in \mathbb R\,,
\end{eqnarray*}
we see that all the conditions of Theorem \ref{theo:6} are fulfilled with
$\lambda^*=-3, $ if
$(I-P)f=0, f \neq 0, $
i.e. if $f \in \ker (L-\lambda^* I)\setminus \{0\}$. Hence, the boundary
value problem
written above
has no solution if
$f(x,t) =
\sin x
\, \sin 2t$
and if $\alpha , \beta \in [-5+\varepsilon, -\varepsilon], $ for some
$\varepsilon>0$.


\begin{thebibliography}{00}

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spectrum and existence of solutions for equations with asymmetric
nonlinearities, to
appear in Proc. Roy. Soc. Edinburgh Sect. A.

\bibitem{dan} E. Dancer, On the Dirichlet problem for weakly nonlinear
elliptic partial differential equations,
Proc. Roy. Soc. Edinburgh Sect. A, 76A, (1977), 283-300.

\bibitem{fab} C. Fabry, Inequalities Verified by Asymmetric Nonlinear
Operators,
Nonlinear Anal. TMA 33 (1998), 121-137.


\bibitem{fa-fo} C. Fabry, A. Fonda, Nonlinear resonance in asymmetric
oscillators,
J. Differential Equations, 147 (1998), 58-78.


\bibitem{fu} S. Fu\v c\'\i k, Boundary value problems with jumping
nonlinearities, \v Casopis Pest. Mat.
101 (1976), 69-87.

\bibitem{fu2} S. Fu\v c\'\i k, A. Kufner, Nonlinear Differential Equations,
Elsevier Science, The Netherlands,
1980.

\bibitem{gak} Th. Gallou\"et,  O. Kavian,
Resonance for jumping non-linearities,
Commun. Partial Differ. Equations 7 (1982), 325-342 .

\bibitem{Kato} T. Kato, Perturbation theory for linear operators,
Springer-Verlag, Berlin, 1966.

\bibitem{lm1}
A. C. Lazer and P. J. McKenna, Existence,
uniqueness, and stability of oscillations in differential equations
with asymmetric nonlinearities, Trans. Amer. Math. Soc., 315 (1989), 721-739.

\bibitem{kra} A.M. Krasnosel'skii, On bifurcation points of equations with
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\end{thebibliography}  \bigbreak

\noindent {\sc A. K. Ben-Naoum} \\
Universit\'e de Mons - Hainaut  \\
Institut de math\'ematique et d'informatique, \\
6 Avenue du Champ de Mars, B - 7000 Mons,  Belgium \\
e-mail: kbn@bsb.be \smallskip

\noindent{\sc C. Fabry}  \\
Universit\'e catholique de Louvain\\
D\'epartement de math\'ematique \\
2 Chemin du cyclotron, B - 1348 Louvain-la-Neuve, Belgium \\
e-mail: fabry@amm.ucl.ac.be \smallskip

\noindent{\sc D. Smets} \\
Universit\'e catholique de Louvain\\
D\'epartement de math\'ematique \\
2 Chemin du cyclotron, B - 1348 Louvain-la-Neuve, Belgium \\
e-mail: smets@amm.ucl.ac.be
  \end{document}

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