\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
2007 Conference on Variational and Topological Methods: Theory, Applications,
Numerical Simulations, and Open Problems.
{\em Electronic Journal of Differential Equations},
Conference 18 (2010),  pp. 33--44.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document} \setcounter{page}{33}
\title[\hfilneg EJDE-2010/Conf/18/\hfil The Fu\v{c}\'{\i}k
spectra for multi-point BVPs]
{The Fu\v{c}\'{\i}k spectra for multi-point boundary-value
problems}

\author[G. Holubov\'{a}, P. Ne\v{c}esal\hfil EJDE/Conf/18 \hfilneg]
{Gabriela Holubov\'{a}, Petr Ne\v{c}esal}  % in alphabetical order

\address{Gabriela Holubov\'{a}\newline
University of West Bohemia, Univerzitn\'{\i} 22,
306 14 Plze\v{n}, Czech Republic}
\email{gabriela@kma.zcu.cz}

\address{Petr Ne\v{c}esal\newline
University of West Bohemia, Univerzitn\'{\i} 22,
306 14 Plze\v{n}, Czech Republic}
\email{pnecesal@kma.zcu.cz}

\thanks{Published July 10, 2010.}
\subjclass[2000]{34B10, 34B15, 34L05}
\keywords{Fu\v{c}\'{\i}k spectrum; asymmetric nonlinearities;
\hfill\break\indent 
multi-point boundary value problem; suspension bridge with two towers}

\begin{abstract}
We study the structure of the Fu\v{c}\'{\i}k spectra
for the linear multi-point differential operators.
We introduce a variational approach in order to obtain a robust and
global algorithm which is suitable for the exploration of unknown
Fu\v{c}\'{\i}k spectrum structure. We apply our approach in the case of
the four-point selfadjoint differential operator of the fourth order
which is closely connected to the nonlinear model of a suspension bridge
with two towers. Moreover, we reconstruct the Fu\v{c}\'{\i}k spectra
in the case of four-point non-selfadjoint ordinary
differential operators of the second order in order to demonstrate
their non-trivial and interesting structure.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\definecolor{seda}{rgb}{0.70, 0.70, 0.70}

\section{Introduction}

In this paper, we investigate the structure of the
{Fu\v{c}\'{\i}k} spectrum
\[
\Sigma(L) = \{(\alpha,\beta)\in {\mathbb{R}}^{2}:
 Lu = \alpha u^{+} - \beta u^{-}
\text{ has a nontrivial solution}\},
\]
where $L$ is the linear operator and $u^{+} := \max\{u, 0\}$,
$u^{-} := \max\{-u, 0\}$. The {Fu\v{c}\'{\i}k} spectra of various differential operators have
been investigated by many authors (see \cite{camdan, horrei, krejci, marmar, necesal2, necesal, schechter}
and the references therein) and the most of the results are proved for
selfadjoint operators.
The selfadjointness allows to use variational approach and essentially influences the structure of
the {Fu\v{c}\'{\i}k} spectrum (see results for general linear selfadjoint operator by Ben-Naoum, Fabry and
Smets in \cite{NFS}). Motivated by \cite{holnec}, we focus on the multi-point differential operators,
especially on the non-selfadjoint operators, in order to demonstrate how the non-selfadjointness
results in non-standard structures of the {Fu\v{c}\'{\i}k} spectrum which are not typical for
ordinary differential operators.

The paper is organized as follows. Section \ref{sekce2} is devoted
to the selfadjoint
multi-point differential operator of the fourth order ${L^{\mathtt{S}}}$. The corresponding
{Fu\v{c}\'{\i}k} spectrum $\Sigma({L^{\mathtt{S}}})$ is defined in a weak sense and using the embedding theorems,
we prove the regularity result for weak solutions. Moreover, due to the selfadjointness of ${L^{\mathtt{S}}}$,
we design an algorithm based on the variational approach
in order to explore the structure of the {Fu\v{c}\'{\i}k} spectrum $\Sigma({L^{\mathtt{S}}})$.
In Section \ref{sekce3}, the main results of \cite{holnec} are briefly recalled
and the complete and precise analytical description of the {Fu\v{c}\'{\i}k} spectrum is provided
for the non-selfadjoint four-point differential operator ${L}$.
The structure of $\Sigma({L})$ exhibits the non-standard and interesting phenomena
which are not obvious for ordinary differential operators:
monotonicity and smoothness of the {Fu\v{c}\'{\i}k}~branches are lost,
the {Fu\v{c}\'{\i}k}~branches intersect away from the diagonal
and the nodal properties of the corresponding {Fu\v{c}\'{\i}k}~eigenfunctions are not preserved.
The last Section \ref{sekce4} concerns the adjoint operator ${L^{*}}$ of ${L}$ and
reveals the correspondence between the {Fu\v{c}\'{\i}k} spectra $\Sigma({L^{*}})$ and
$\Sigma({L})$.

In this paper, let us adopt the concept of the multi-point differential operators introduced in \cite{locker}.
Thus, suppose that ${\mathcal{P}} = \{a = x_{0} < x_{1} < \dots < x_{m} = b\}$ is a given partition
of the interval $[a, b]$. Let $H^{n}({\mathcal{P}})$ denote the set of all functions $u\in L^{2}(a,b)$
with the following two properties
\begin{enumerate}
\item On each subinterval $[x_{i-1}, x_{i}]$, the function $u = u(x)$ possesses both right-hand
  and left-hand limits at the endpoints $x_{i-1}$ and $x_{i}$, respectively.
  For $i = 1, \dots, m$, let $u_{i}:[x_{i-1}, x_{i}] \to {\mathbb{R}}$ be the function defined by
  $u_{i} \equiv u$ on $(x_{i-1}, x_{i})$, $u(x_{i-1}) = u(x_{i-1}+)$ and $u(x_{i}) = u(x_{i}-)$.
  The functions $u_{1}, \dots,$ $u_{m}$ are called the \emph{components} of $u$, which is denoted
  by $u = (u_{1}, \dots, u_{m})$.
\item The components $u_{i}$ belong to $H^{n}([x_{i-1}, x_{i}])$ for $i = 1, \dots, m$.
\end{enumerate}

Finally, let us note that in the next sections,
we follow mainly the paper \cite{holnec} and using three different approaches,
we recover the non-trivial {Fu\v{c}\'{\i}k} spectrum structures of the three multi-point operators
${L^{\mathtt{S}}}$, ${L}$ and ${L^{*}}$.


\section{Selfadjoint multi-point operator}
\label{sekce2}

Let us consider the  multi-point boundary-value problem
\begin{equation}
\begin{gathered}
  - u^{\mathrm{IV}}(x) + \alpha u^{+}(x) - \beta u^{-}(x) = 0, \quad
 x \in  (0, \xi)\cup(\xi, \eta)\cup(\eta, \pi),\\
  u'(0) = u(0) = u(\xi) = u(\eta) = u(\pi) = u'(\pi) = 0,
\quad  0 < \xi < \eta < \pi,
\end{gathered}
\label{klasickaformulace}
\end{equation}
which represents the simplified mathematical model of a suspension bridge with two towers (see Figure \ref{obr1}).
The roadbed of the bridge is modelled as a clamped beam which is suspended by two systems of one-sided springs
with stiffnesses $\alpha$ and $\beta$. The function $u = u(x)$ describes the steady-state displacement
of the roadbed and is measured as positive in downward direction.
Let us recall the Lazer-McKenna normalized suspension bridge model (see \cite{lazmck} or \cite{mckwal})
and also the open problem in \cite{dranec} concerning the explicit form of the resonance set for
such a mathematical model.

Let us define the partition of the interval $[0, \pi]$ as
${\mathcal{P}} = \{0, \xi, \eta, \pi\}$
and let us define the following multi-point boundary values
\[
\begin{array}{lll}
  B_{1}(u) = u (0+)    = u_{1} (0),    & B_{4}(u) = u (\xi+)  = u_{2} (\xi),  & B_{6}(u) = u (\eta+) = u_{3} (\eta), \\
  B_{2}(u) = u'(0+)    = u_{1}'(0),    & B_{5}(u) = u (\eta-) = u_{2} (\eta), & B_{7}(u) = u (\pi-)  = u_{3} (\pi), \\
  B_{3}(u) = u (\xi-)  = u_{1} (\xi),  &                                     & B_{8}(u) = u'(\pi-)  = u_{3}'(\pi),
\end{array}
\]
(recall that $u = (u_{1}, u_{2}, u_{3})$). Let ${L^{\mathtt{S}}} : {\mathrm{dom}}({L^{\mathtt{S}}}) \subset L^{2}(0, \pi) \to L^{2}(0, \pi)$
be the linear multi-point differential operator defined by
\[
{L^{\mathtt{S}}} u(x) := u^{\mathrm{IV}}(x),\quad
{\mathrm{dom}}({L^{\mathtt{S}}}) := \left\{ u \in H^{4}({\mathcal{P}}): B_{i}(u) = 0,\ i = 1, \dots, 8\right\}.
\]
Using Theorem 2 in \cite{locker}, it is straightforward to verify that ${L^{\mathtt{S}}}$ is the selfadjoint operator.
Due to selfadjointness of ${L^{\mathtt{S}}}$, results by Ben-Naoum, Fabry and Smets in \cite{NFS}
concerning general linear selfadjoint operator
can be applied in order to describe the {Fu\v{c}\'{\i}k} spectrum $\Sigma({L^{\mathtt{S}}})$.
Let us remark that in the case of the Navier and the Dirichlet operators of the fourth order,
the qualitative properties of the {Fu\v{c}\'{\i}k} spectrum are investigated in \cite{krejci} and \cite{camdan}.

\begin{figure}[ht]
\begin{center}
  \setlength{\unitlength}{1.08mm}
  \begin{picture}(114, 39)(0, -3)
    \includegraphics[width=12.42cm]{fig1} %{HOL_NEC_fig_1.eps}
    \put(-72,34){$\eta$}
    \put(-100,6){$\xi$}
    \put(-58.5,-2){$\pi$}
    \put(-103,15){$x$}
    \put(-111,25){$u$}
    \put(-64.5,14.5){$u(x)$}
    \put(-43,24){$\alpha$}
    \put(-43,6){$\beta$}
  \end{picture}
\end{center}
\caption{The model of a suspension bridge with two towers}
\label{obr1}
\end{figure}

To introduce the weak formulation of the main problem \eqref{klasickaformulace},
let us define the space of test functions as
$V := \{ v \in H^{2}({\mathcal{P}}): B_{i}(u) = 0,\ i = 1, \dots, 8\}$
and denote by $(u, v)$ the scalar product on $L^{2}(0, \pi)$.

\begin{definition} \rm
Let us define a \emph{weak solution} of \eqref{klasickaformulace} as
a function $u \in V$ such that
\begin{equation}
  (u'', v'') = (\alpha u^{+}(x) - \beta u^{-}(x), v) \quad
 \forall\, v\in V.
  \label{slabaformulace}
\end{equation}
\end{definition}
The following lemma concerns the regularity result for weak solutions
of \eqref{klasickaformulace}.

\begin{lemma} \label{regularita}
If $u$ is a weak solution of \eqref{klasickaformulace} then $u$ is
the classical solution.
Moreover,
\[
u \in C^{2}([0, \pi]),
\quad
u_{1} \in C^{4}([0, \xi]),\
u_{2} \in C^{4}([\xi, \eta]),\
u_{3} \in C^{4}([\eta, \pi]),
\]
where $u = (u_{1}, u_{2}, u_{3})$.
\end{lemma}

\begin{proof}
Let $u\in V$ is a weak solution of \eqref{klasickaformulace},
\begin{equation}
  \int_{0}^{\pi} u''(x) v''(x) \,{\mathrm{d}} x
- \int_{0}^{\pi} g(u(x)) v(x) \,{\mathrm{d}} x = 0 \quad
\forall v\in V,
  \label{hlavnivztah}
\end{equation}
where $g(u) := \alpha u^{+} - \beta u^{-}$.
First of all, since $u\in H^{2}(0, \pi)$, we have that
$u\in C^{1}([0, \pi])$
due to the compact imbedding
$H^{2}(0, \pi){{{\ooalign{\mathhexbox21A\kern.15em\crcr\hfil\lower0.33ex\hbox{\tiny$\succ$}}}}{{\ooalign{\mathhexbox21A\kern.15em\crcr\hfil\lower0.33ex\hbox{\tiny$\succ$}}}}} C^{1}([0, \pi])$.

If we define $U \subset V$ as
\begin{equation}
  U := \{ v \in H^{2}({\mathcal{P}}): v_{1} \equiv 0,\ v_{2}
\in C_{0}^{\infty}(\xi, \eta),\ v_{3} \equiv 0 \},
  \label{prostorU}
\end{equation}
we obtain, using \eqref{hlavnivztah}, that
\begin{equation}
  \int_{\xi}^{\eta} u_{2}''(x) v_{2}''(x) \,{\mathrm{d}} x - \int_{\xi}^{\eta} g(u_{2}(x)) v_{2}(x) \,{\mathrm{d}} x = 0 \quad
  \forall v_{2}\in C_{0}^{\infty}(\xi, \eta).
  \label{poorezu}
\end{equation}
If we integrate by parts two times the second integral in \eqref{poorezu}, we obtain
\begin{equation}
  \int_{\xi}^{\eta} M(x) v_{2}''(x) \,{\mathrm{d}} x = 0 \quad \forall v_{2}\in C_{0}^{\infty}(\xi, \eta),
  \label{poperpartes}
\end{equation}
where $M(x) := u_{2}''(x) - \int_{\xi}^{x}\int_{\xi}^{t} g(u_{2}(\tau)) {\mathrm{d}} \tau {\mathrm{d}} t$.
The equality in \eqref{poperpartes} can be interpreted as $\int_{\xi}^{\eta} M_{2}(x) v_{2}(x) \,{\mathrm{d}} x = 0$,
where $M_{2}(x)$ denotes the second derivative of $M(x)$ in the sense of distributions.
Using the generalized variational lemma, we obtain that $M_{2}(x) = 0$ in the space of distributions,
which means that there exists the constants $c_{0},c_{1}\in{\mathbb{R}}$
such that
\begin{equation}
  M(x) = c_{1} x + c_{0} \quad \text{for almost all } x\in(\xi,\eta).
  \label{linearno}
\end{equation}
Now, let us define the function $F:[\xi, \eta]\times{\mathbb{R}}\to{\mathbb{R}}$ by
\[
  F(x, z) = z - \int_{\xi}^{x}\!\!\!\int_{\xi}^{t} g(u_{2}(\tau))
{\mathrm{d}} \tau {\mathrm{d}} t - c_{1} x - c_{0}.
\]
For every $x\in[\xi, \eta]$ there exists exactly one $z\in{\mathbb{R}}$ such that $F(x, z) = 0$.
Due to $u_{2} \in C^{1}([\xi, \eta])$ and $g$ is the continuous function, the function $F(x,z)$
has continuous partial derivatives of the first order on $[\xi, \eta]\times{\mathbb{R}}$.
Thus, using Implicit Theorem, we obtain that the function $z = z(x)$ given implicitely
by $F(x, z(x)) = 0$ is continuous together with its first derivative on $[\xi, \eta]$.
If we recall the definition of $M(x)$, \eqref{linearno} can be written as
$F(x, u_{2}''(x)) = 0$ for almost all $x\in(\xi, \eta)$, which implies that
$z(x) = u_{2}''(x)$ for almost all $x\in(\xi, \eta)$. The continuity of $z(x)$ on $[\xi, \eta]$
immediately gives us that $z(x) = u_{2}''(x)$ for every $x\in[\xi, \eta]$
(note that for all $x\in[\xi, \eta]: u_{2}'(x) = u_{2}'(\xi) + \int_{\xi}^{\eta} u_{2}''(t)\,{\mathrm{d}} t =
u_{2}'(\xi) + \int_{\xi}^{\eta}z(t)\,{\mathrm{d}} t$).
Moreover, due to $z\in C^{1}([\xi, \eta])$, we conclude that $u_{2}\in C^{3}([\xi, \eta])$.

Now, using integration by parts to both integrals in \eqref{poorezu}, we obtain
\[
  \int_{\xi}^{\eta} \Big[ u_{2}'''(x) - \int_{\xi}^{x} g(u_{2}(t))
 \,{\mathrm{d}} t\Big] v_{2}'(x)\,{\mathrm{d}} x = 0 \quad
  \forall v_{2}\in C_{0}^{\infty}(\xi, \eta).
\]
If we proceed in the analog way as in the case of \eqref{poperpartes}, we obtain that $u_{2}\in C^{4}([\xi, \eta])$.
This enables us to integrate by parts two times the first integral in \eqref{poorezu} to get
\[
  \int_{\xi}^{\eta} \left[ u_{2}^{\mathrm{IV}}(x) - g(u_{2}(x)) \right]
  v_{2}(x) \,{\mathrm{d}} x = 0 \quad
  \forall v_{2}\in C_{0}^{\infty}(\xi, \eta),
\]
which implies that $u_{2}^{\mathrm{IV}}(x) = g(u_{2}(x))$ for
almost all $x \in (\xi, \eta)$ due to the
density of $C_{0}^{\infty}(\xi, \eta)$ in $L^{2}(\xi, \eta)$.
The continuity of $g$ and $u_{2}^{\mathrm{IV}}$
ensures that $u_{2}^{\mathrm{IV}}(x) = g(u_{2}(x))$ for every
$x \in [\xi, \eta]$.

Finally, if we replace the definition of $U$ in \eqref{prostorU} as
\[
  U := \{ v \in H^{2}({\mathcal{P}}): v_{1}
\in C_{0}^{\infty}(0, \xi),\ v_{2} \equiv 0,\ v_{3} \equiv 0 \},
\]
or as
\[
  U := \{ v \in H^{2}({\mathcal{P}}): v_{1} \equiv 0,\ v_{2}
\equiv 0,\ v_{3} \in C_{0}^{\infty}(\eta, \pi) \},
\]
we obtain in the analog way that $u^{\mathrm{IV}}(x) = g(u(x))$
for all $x \in (0, \xi) \cup (\eta,\pi)$.
Thus, $u$ is the classical solution of \eqref{klasickaformulace}.
Moreover, if we integrate by parts two times all three integrals in
$\int_{0}^{\xi} u_{1}'' v_{1}''\,{\mathrm{d}} x + \int_{\xi}^{\eta} u_{2}'' v_{2}''\,{\mathrm{d}} x + \int_{\eta}^{\pi} u_{3}'' v_{3}''\,{\mathrm{d}} x$
then \eqref{hlavnivztah} can be written in the following form
\[
  \left[u_{1}''(\xi) - u_{2}''(\xi)\right] v'(\xi) +
  \left[u_{2}''(\eta) - u_{3}''(\eta)\right] v'(\eta) = 0 \quad \forall v\in V,
\]
which implies that $u''(\xi-) = u''(\xi+)$ and $u''(\eta-) = u''(\eta+)$. Thus, we conclude that
$u\in C^{2}([0, \pi])$.
\end{proof}

\begin{figure}[ht]
\begin{center}
  \begin{minipage}{6.25cm}
    \setlength{\unitlength}{0.78mm}
    \begin{picture}(80, 80)(0, 0)
      \put(0,0){\includegraphics[width=6.25cm]{fig2}} %{HOL_NEC_fig_2.eps}}
      \put(58,-2){\makebox(0,0)[cm]{\scriptsize$\sqrt{\alpha}$}}
      \put(-2,57){\makebox(0,0)[cm]{\scriptsize$\sqrt{\beta}$}}
    \end{picture}
  \end{minipage}
  \begin{tabular}{cc}
    \scriptsize eigenfunctions & \scriptsize the {Fu\v{c}\'{\i}k}~eigenfunctions\\
      \includegraphics[width=2.6cm]{fig2aa} 
    & \includegraphics[width=2.6cm]{fig2ab} \\
      \includegraphics[width=2.6cm]{fig2ba} 
    & \includegraphics[width=2.6cm]{fig2bb} \\
      \includegraphics[width=2.6cm]{fig2ca} 
    & \includegraphics[width=2.6cm]{fig2cb} \\
      \includegraphics[width=2.6cm]{fig2da} 
    & \includegraphics[width=2.6cm]{fig2db} \\
      \includegraphics[width=2.6cm]{fig2ea} 
    & \includegraphics[width=2.6cm]{fig2eb} \\
      \includegraphics[width=2.6cm]{fig2fa} 
    & \includegraphics[width=2.6cm]{fig2fb} \\
      \includegraphics[width=2.6cm]{fig2ga} 
    & \includegraphics[width=2.6cm]{fig2gb}
  \end{tabular}
\end{center}
\caption{The approximation of $\Sigma({L^{\mathtt{S}}})$ for $\xi = \frac{5\pi}{12} < \frac{7\pi}{12} = \eta$
         and the corresponding {Fu\v{c}\'{\i}k}~eigenfunctions.}
\label{obr_fucik_Lmost}
\end{figure}

To explore the {Fu\v{c}\'{\i}k} spectrum for the problem \eqref{klasickaformulace}, let us
focus first on the corresponding eigenvalue problem in the case of $\alpha = \beta$.
Let us consider the space $V$ as the Hilbert space
with the inner product ${(u, v)_{V}} := \int_{0}^{\pi} u''(x) v''(x)\,{\mathrm{d}} x$ and
the norm ${\left\|{u}\right\|_{V}}:= \sqrt{{\left(u, u\right)_{V}}}$
and let us define the operator $A: V \to V$ as
\[
{\left({A(u)}, {v}\right)_{V}} := \int_{0}^{\pi} u(x) v(x) \, {\mathrm{d}} x
\]
for all $u, v\in V$. Thus, in the case of $\alpha = \beta =: \lambda$, the weak formulation
of the problem \eqref{klasickaformulace} has the form of the following \emph{weak eigenvalue problem}
\[
\mu u = A (u),
\]
where $\mu = \frac{1}{\lambda}$.
The operator $A$ is positive, self-adjoint and compact
(recall the compact embedding $H^{2}(0,\pi) {{{\ooalign{\mathhexbox21A\kern.15em\crcr\hfil\lower0.33ex\hbox{\tiny$\succ$}}}}{{\ooalign{\mathhexbox21A\kern.15em\crcr\hfil\lower0.33ex\hbox{\tiny$\succ$}}}}} L^{2}(0,\pi)$).
Thus, using Courant-Fisher Principle, we obtain that
$A$ has a countable set of \emph{weak eigenvalues} $\{\mu_{k}\}$ of the finite multiplicity such that
$\mu_{1} \ge \mu_{2} \ge \mu_{3} \ge \dots > 0$ and $\lim\limits_{k\to +\infty}\mu_{k} = 0$.
Hence, using $\lambda = \frac{1}{\mu}$, we obtain the non-decreasing sequence $\{\lambda_{n}\}$ such that
\[
0 < \lambda_{1} \le \lambda_{2} \le \lambda_{3}
 \le \dots,\quad \lim_{n\to +\infty}\lambda_{n} = +\infty,
\]
and that
\[
\int_{0}^{\pi} \varphi_{n}''(x) v''(x)\,{\mathrm{d}} x
= \lambda_{n} \int_{0}^{\pi} \varphi_{n}(x) v(x) \, {\mathrm{d}} x
\quad \forall v\in V,
\]
where the \emph{weak eigenfunction} $\varphi_{n}\in V$ satisfies
$\mu_{n} \varphi_{n} = A (\varphi_{n})$.
Finally, using Lemma \ref{regularita},
we conclude that $\varphi_{n}$ is the nontrivial classical solution
of \eqref{klasickaformulace}
with $\alpha = \beta = \lambda_{n}$, $n\in{\mathbb{N}}$.
Thus, we have
\[
\forall n \in {\mathbb{N}}: (\lambda_{n}, \lambda_{n}) \in \Sigma({L^{\mathtt{S}}}).
\]

Now, let us adopt the variational approach introduced in \cite{necesal} in order to explore numericaly
the points of the {Fu\v{c}\'{\i}k} spectrum $\Sigma({L^{\mathtt{S}}})$.
Let $\mu\in{\mathbb{R}}\setminus\sigma({L^{\mathtt{S}}})$ and $\delta\in{\mathbb{R}}$ be such that $(\mu+\delta)\not\in\sigma({L^{\mathtt{S}}})$,
where $\sigma({L^{\mathtt{S}}})$ denotes the spectrum of ${L^{\mathtt{S}}}$.
If we take into account the  transformation
\begin{gather*}
\mathcal{T}_{\mu,\delta}   = \mathcal{T}_{\mu,\delta}
   (\alpha, \beta, u) = (m, \tilde{\lambda}, v), \quad
\mathcal{T}^{-1}_{\mu,\delta} = \mathcal{T}^{-1}_{\mu,\delta}
(m, \tilde{\lambda}, v) = (\alpha, \beta, u),
\\
\mathcal{T}_{\mu,\delta} :
\begin{cases}
   m = \frac{\beta - \alpha}{\beta + \alpha - 2\mu},\\
   \tilde{\lambda}=\frac{2\mu - \alpha - \beta}
{2(\mu - \alpha)(\mu - \beta) {+\delta(2\mu - \alpha - \beta)}},\\
  v = \left(\mu I - {L^{\mathtt{S}}}\right) u,
\end{cases}
\quad
\mathcal{T}^{-1}_{\mu,\delta} :
\begin{cases}
   \alpha = \mu - \frac{1{-\delta\tilde{\lambda}}}
 {\tilde{\lambda}(1 + m)},\\
   \beta = \mu - \frac{1{-\delta\tilde{\lambda}}}{\tilde{\lambda}(1 - m)},\\
  u = \left(\mu I - {L^{\mathtt{S}}}\right)^{-1} v,
\end{cases}
\end{gather*}
then the {Fu\v{c}\'{\i}k} spectrum problem
${L^{\mathtt{S}}} u = \alpha u^{+} - \beta u^{-}$
reads as the nonlinear problem
\begin{equation}
\left((\mu+\delta) I - {L^{\mathtt{S}}}\right)^{-1} v =
\tilde\lambda \left(v + m \left(I - \delta\left[(\mu+\delta) I - {L^{\mathtt{S}}}\right]^{-1}\right)|v|\right).
\label{eigen2}
\end{equation}
Due to homogenity of the nonlinear eigenpair problem \eqref{eigen2},
the critical points of the corresponding Rayleigh quotient
$J:L^{2}(0, \pi)\to{\mathbb{R}}$
\begin{gather*}
J(v) = \frac{F(v)}{G(v)},\quad
F(v) = \frac{1}{2}\int_{0}^{\pi}\left((\mu+\delta) I - {L^{\mathtt{S}}}\right)^{-1} v \cdot v\,\mathrm{d}x,\\[6pt]
G(v) = \frac{1}{2}\int_{0}^{\pi}v^{2} + m \left(I - \delta\left[(\mu+\delta) I - {L^{\mathtt{S}}}\right]^{-1}\right)|v|v\,\mathrm{d}x,
\end{gather*}
together with their critical values $\tilde\lambda = J(v)$
are in one to one correspondence with
eigenpairs $(\tilde\lambda, v)$ of \eqref{eigen2}.

In order to simplify the following notation, let us consider the subsequence $\{\lambda_{n_{k}}\}$
such that $0 < \lambda_{n_{1}} < \lambda_{n_{2}} < \lambda_{n_{3}} < \dots$ and that
$\bigcup_{n=1}^{+\infty}\{\lambda_{n}\} = \bigcup_{k=1}^{+\infty}\{\lambda_{n_{k}}\}$
and denote it again by $\{\lambda_{n}\}$.
First, let us note that for $m = 0$, the equation \eqref{eigen2} can be written in the form
${L^{\mathtt{S}}} v = (\mu + \delta - \frac{1}{\tilde\lambda})v$.
Thus, if we set $\mu < 0$ and $\delta = 0$ then we can characterize the first eigenvalue of ${L^{\mathtt{S}}}$
as $\lambda_{1} = \mu - \frac{1}{\tilde\lambda_{1}}$, where $\tilde\lambda_{1} = J(\psi_{1})$
with $\psi_{1} = \arg\min J(v)$. The eigenfunction $\varphi_{1}$ corresponding to $\lambda_{1}$
is given by $\varphi_{1} = \left(\mu I - {L^{\mathtt{S}}}\right)^{-1} \psi_{1}$.
Moreover, if we take $\mu < 0$ and $\delta > 0$ such that $(\mu+\delta)\in(\lambda_{1}, \lambda_{2})$ then
$\lambda_{1} = \mu + \delta - \frac{1}{\max J(v)}$ and
$\lambda_{2} = \mu + \delta - \frac{1}{\min J(v)}$.

Second, in order to obtain the approximation of $\Sigma({L^{\mathtt{S}}})$ depicted in Figure \ref{obr_fucik_Lmost},
let us proceed in the following way.
Let us fix $\mu < 0$, $\delta = \frac{\lambda_{i} + \lambda_{i+1}}{2} - \mu$, $i\in{\mathbb{N}}$, and
$m \in [-1, 1]$. Then the minimization and maximization process of the functional $J$
gives us two critical values $\tilde\lambda = J(v)$ and the inverse transformation
$\mathcal{T}^{-1}_{\mu,\delta} (m, \tilde{\lambda}, v) = (\alpha, \beta, u)$ recovers two points
of the {Fu\v{c}\'{\i}k} spectrum $\Sigma({L^{\mathtt{S}}})$.

\begin{remark} \rm
To reduce the problem of finding the critical points of the functional
$J : L^{2}(0, \pi) \to {\mathbb{R}}$ onto a finite-dimensional space,
we consider the following descretization. Denote $h := \frac{\pi}{n}$, $n\in{\mathbb{N}}$, and let
$x_{i} := i h$, $i = 0, 1, \dots, n$, be the equidistant mesh of the interval $[0, \pi]$.
Moreover, let $\xi = x_{p_{1}} < x_{p_{2}} = \eta$ with $0 < p_{1} < p_{2} < n$
and denote by $U_{i}$ the approximation of $u(x_{i})$, $i = 0, 1, \dots, n$.
Since $U_{0} = U_{p_{1}} = U_{p_{2}} = U_{n} = 0$, we take
$$U := [U_{1}, U_{2}, \dots, U_{p_{1}-1}, U_{p_{1}+1}, \dots, U_{p_{2}-1}, U_{p_{2}+1}, \dots, U_{n-1}]$$
as the approximation of $u$ on the interval $[0, \pi]$.
Finally, the approximation of the selfadjoint operator ${L^{\mathtt{S}}}$
has the form of the following five-diagonal symmetric matrix of order $n-3$
{\scriptsize
\begin{gather*}
p_{1}-1 \\
\frac{1}{h^{4}}\cdot
\left\lceil
\begin{array}{rrrrrrrrrrrrrr}
 7 & -4 &  1 &    &    &    &    & \color{seda} 0 \\
-4 &  6 & -4 &  1 &    &    &    & \color{seda} 0 \\
 1 & -4 &  6 & -4 &  1 &    &    & \color{seda} 0 \\
   & \ddots & \ddots & \ddots & \ddots & \ddots & & \vdots \\
   &    &  1 & -4 &  6 & -4 &  1 & \color{seda} 0 \\
   &    &    &  1 & -4 &  6 & -4 &  1 \\
   &    &    &    &  1 & -4 &  6 & -4 &  \color{seda} 0 \\
   &    &    &    &    &  1 & -4 &  6 &  1 & \color{seda} 0 & \color{seda} 0 & \color{seda} 0 & \color{seda} 0 & \dots \\
   &    &    &    &    &    &  \color{seda} 0 &  1 &  6 & -4 &  1 \\
   &    &    &    &    &    &    &  \color{seda} 0 & -4 &  6 & -4 &  1 \\
   &    &    &    &    &    &    &    &  1 & -4 &  6 & -4 &  1 \\
   &    &    &    &    &    &    &    &    & \ddots & \ddots & \ddots & \ddots & \ddots
\end{array}
\right\rceil
. 
\,
\begin{array}{l}
\\[14pt]
p_{1}-1
\end{array}
\end{gather*}
}
\end{remark}

To conclude this section, let us formulate the following conjecture
based on the numerical experiments
(see Figure \ref{obr_fucik_Lmost}).

\begin{conjecture} $\,$
\begin{enumerate}
\item Each point $E_{n} := (\lambda_{n}, \lambda_{n})$, $n \in {\mathbb{N}}$,
      gives arise to exactly one or two {Fu\v{c}\'{\i}k}~curves of $\Sigma({L^{\mathtt{S}}})$.
\item Any {Fu\v{c}\'{\i}k}~curve that goes through the point $E_{n}$ with $n \ge 3$ can be described
      by a strictly decreasing function $\alpha \mapsto \beta(\alpha)$ which is not necessarily convex.
      Compare it with the {Fu\v{c}\'{\i}k} spectrum for the Dirichlet operator of the second order in \cite{drabek}
      which is given by explicit analytic formulas and the nontrivial {Fu\v{c}\'{\i}k}~curves are described
      as strictly decreasing convex functions.
\item In the case of the symmetrical settings of $\xi$ and $\eta$
      with respect to the midpoint of the interval $[0, \pi]$,
      we have for every eigenfunction $\varphi_{n}$ that
      $\varphi_{n}(x+\frac{\pi}{2})$ is an even or an odd function.
      But, this statement is not necessarily valid for the {Fu\v{c}\'{\i}k}~eigenfunctions when $\alpha \neq \beta$
      (see Figure \ref{obr_fucik_Lmost} with the emphasize to the points $P_2$, $P_{4}$ and $P_{6}$).
\end{enumerate}
\end{conjecture}

\section{Non-selfadjoint multi-point operator}
\label{sekce3}

In this section, let us describe the {Fu\v{c}\'{\i}k} spectrum of the
following four-point boundary value problem
\begin{equation}
\begin{gathered}
  u''(x) + \alpha u^{+}(x) - \beta u^{-}(x) = 0, \quad  x\in (0, \pi),\\
  u'(0) = u'(\xi),\quad u(\pi) = u(\eta),        \quad \xi\in (0, \pi),\ \eta\in (0, \pi),
\end{gathered}
\label{odeuloha}
\end{equation}
where we assume in addition that $\xi > \frac{\pi + \eta}{2}$ in
order to simplify the following notation.
Let us denote by ${\mathcal{P}} := \{0, \eta, \xi, \pi\}$ the
partition of the interval $[0, \pi]$
and let us define the following multi-point boundary values
(note that $u = (u_{1}, u_{2}, u_{3})$)
\[
\begin{array}{lll}
  B_{1}(u):=u_{2}(\xi)  - u_{3}(\xi),  & B_{3}(u):=u_{2}'(\xi)  - u_{3}'(\xi),  & B_{5}(u):=u_{1}'(0)   - u_{3}'(\xi),\\
  B_{2}(u):=u_{1}(\eta) - u_{2}(\eta), & B_{4}(u):=u_{1}'(\eta) - u_{2}'(\eta), & B_{6}(u):=u_{1}(\eta) - u_{3}(\pi).
\end{array}
\]
Let us associate to \eqref{odeuloha} the linear multi-point
differential operator
${L} : {\mathrm{dom}}({L}) \subset L^{2}(0, \pi) \to L^{2}(0, \pi)$
defined by
\[
{L} u(x) := -u''(x),\quad
{\mathrm{dom}}({L}) := \left\{ u \in H^{2}({\mathcal{P}}): B_{i}(u) = 0,
\ i=1,\dots,6 \right\}.
\]
Using Theorem 2 in \cite{locker}, it is possible to verify that
the operator ${L}$ is non-selfadjoint operator.

The complete description of the {Fu\v{c}\'{\i}k} spectrum
$\Sigma({L})$ is provided in \cite{holnec},
thus, in the following text, let us only recall the main results.
Let us note that the non-selfadjointness of the operator ${L}$
results in nontrivial structure of the
{Fu\v{c}\'{\i}k} spectrum with interesting patterns (see Figure \ref{obr_FSP_fourp}).

\begin{figure}[ht]
\begin{center}
  \setlength{\unitlength}{0.78mm}
  \begin{picture}(152.7, 50)(0, 0)
    \put(  0, 0){\includegraphics[width=3.9cm]{fig3a}} %{HOL_NEC_fig_3a.eps}}
    \put( 52, 0){\includegraphics[width=3.9cm]{fig3b}} %{HOL_NEC_fig_3b.eps}}
    \put(104, 0){\includegraphics[width=3.9cm]{fig3c}} %{HOL_NEC_fig_3c.eps}}
    \put(140,-3){\makebox(0,0)[cm]{\scriptsize$\sqrt{\alpha}$}}
    \put( -4,40){\makebox(0,0)[cm]{\scriptsize$\sqrt{\beta}$}}
  \end{picture}
\end{center}
\caption{The decomposition of the {Fu\v{c}\'{\i}k} spectrum $\Sigma({L})$ into
         $\Sigma({L^{ \mathtt{P}{\xi} }})$, $\Sigma({L^{ \mathtt{P}{\eta} }})$ (black curves) and $\Sigma({L^{ \mathtt{3p} }})$ (orange curves)
         for $\xi = \pi - \eta$ (left, middle) and for $\xi \neq \pi - \eta$ (right).}
\label{obr_FSP_fourp}
\end{figure}

Let us define the following multi-point differential operators
\[
{L^{ \mathtt{P}{\xi} }} u := {L^{ \mathtt{P}{\eta} }} u := {L^{ \mathtt{DN} }} u := {L^{ \mathtt{3p} }} u := -u'',
\]
\[
\begin{array}{rcl}
{\mathrm{dom}}({L^{ \mathtt{P}{\xi} }}) & := & \left\{ u \in H^{2}({\mathcal{P}}): B_{i}(u) = 0,\ i = 1,\dots,4,5,7 \right\}, \\
{\mathrm{dom}}({L^{ \mathtt{P}{\eta} }})& := & \left\{ u \in H^{2}({\mathcal{P}}): B_{i}(u) = 0,\ i = 1,\dots,4,6,8 \right\},
\end{array}
\]
where
\[
B_{7}(u) = u_{1}(0)- u_{3}(\xi),\quad B_{8}(u) = u_{1}'(\eta) - u_{3}'(\pi),
\]
and
\[
\begin{array}{rcl}
{\mathrm{dom}}({L^{ \mathtt{DN} }}) & := & \left\{ u \in H^{2}([0, \pi]):
u\left(\frac{\xi}{2}\right) = u'\left(\frac{\pi+\eta}{2}\right) = 0\right\}, \\
{\mathrm{dom}}({L^{ \mathtt{3p} }}) & := & \left\{ u \in H^{2}([0, \pi]):
u'(0) - u'(\xi) = u'\left(\frac{\pi+\eta}{2}\right) = 0,\ u(0)u(\xi) \le 0\right\}.
\end{array}
\]
According to Theorem 6 in \cite{holnec}, we have that
\begin{equation}
\Sigma({L}) =
\Sigma({L^{ \mathtt{P}{\xi} }}) \cup \Sigma({L^{ \mathtt{P}{\eta} }}) \cup \Sigma({L^{ \mathtt{3p} }}) \neq
\Sigma({L^{ \mathtt{P}{\xi} }}) \cup \Sigma({L^{ \mathtt{P}{\eta} }}) \cup \Sigma({L^{ \mathtt{DN} }})
\label{sekce3_eq_1}
\end{equation}
in contrast to
\[
\sigma({L}) =
\sigma({L^{ \mathtt{P}{\xi} }}) \cup \sigma({L^{ \mathtt{P}{\eta} }}) \cup \sigma({L^{ \mathtt{3p} }}) =
\sigma({L^{ \mathtt{P}{\xi} }}) \cup \sigma({L^{ \mathtt{P}{\eta} }}) \cup \sigma({L^{ \mathtt{DN} }}),
\]
where the spectra $\sigma({L^{ \mathtt{P}{\xi} }})$, $\sigma({L^{ \mathtt{P}{\eta} }})$ and $\sigma({L^{ \mathtt{3p} }}) = \sigma({L^{ \mathtt{DN} }})$
are identified as pure point discrete spectra made only of the real eigenvalues
\[
{\lambda^{ \mathtt{P}{\xi} }_{k}}  := \left(\frac{2 k \pi}{\xi}\right)^{2},\quad
{\lambda^{ \mathtt{P}{\eta} }_{m}} := \left(\frac{2 m \pi}{\pi - \eta}\right)^{2}\quad \text{and} \quad
{\lambda^{ \mathtt{DN} }_{l}}      := \left(\frac{(2 l + 1)\pi}{\pi + \eta - \xi}\right)^{2},\, k,l,m\in{\mathbb{N}}_{0}.
\]
(see \cite{holnec}, \cite{drabek} and \cite{fucik}
for the explicit analytical description of $\Sigma({L^{ \mathtt{P}{\xi} }})$, $\Sigma({L^{ \mathtt{P}{\eta} }})$ and $\Sigma({L^{ \mathtt{DN} }})$
and note that $\Sigma({L^{ \mathtt{P}{\xi} }}) = \Sigma({L^{ \mathtt{P}{\eta} }})$ for $\xi = \pi - \eta$).
Moreover, the {Fu\v{c}\'{\i}k} spectrum $\Sigma({L^{ \mathtt{DN} }})$ determines the intersection of
$\Sigma({L^{ \mathtt{3p} }})$ and $\Sigma({L^{ \mathtt{P}{\xi} }})$
\[
\Sigma({L^{ \mathtt{3p} }}) \cap \Sigma({L^{ \mathtt{P}{\xi} }})
 = \Sigma({L^{ \mathtt{DN} }}) \cap \Sigma({L^{ \mathtt{P}{\xi} }})
\quad
\text{on }{\mathbb{R}}^{+}\times{\mathbb{R}}^{+},
\]
which is illustrated in Figure \ref{obr_FSP_fourp}
for two symmetric and one non-symmetric settings of $\xi$ and $\eta$
with respect to the midpoint of $[0, \pi]$.

Finally, to describe explicitely the {Fu\v{c}\'{\i}k} spectrum 
$\Sigma({L^{ \mathtt{3p} }})$,
let us denote for $k, l \in {\mathbb{N}}_{0}$
\begin{align*}
  {S^{ \mathtt{P\xi} }_{k}} & :=
  \left\{ (\alpha, \beta)\in{\mathbb{R}}^{2} :
  0 \le \frac{\xi}{\pi} - \frac{k}{\sqrt{\alpha}} -
  \frac{k}{\sqrt{\beta}} \le \frac{1}{\sqrt{\alpha}} +
  \frac{1}{\sqrt{\beta}}\right\}\,, \\
{S^{ \mathtt{DN} }_{l}}
 & := \big\{ (\alpha, \beta)\in{\mathbb{R}}^{2} :
  \frac{1}{\max\{\sqrt{\alpha},\sqrt{\beta}\}}  
  \\
  &   \le \frac{\pi + \eta - \xi}{\pi} - \frac{l}{\sqrt{\alpha}} -
  \frac{l}{\sqrt{\beta}} \le \frac{1}{\min\{\sqrt{\alpha},\sqrt{\beta}\}}\big\}.
\end{align*}

\begin{proposition}
\label{sekce3_prop_1}
The {Fu\v{c}\'{\i}k} spectrum of the three-point operator ${L^{ \mathtt{3p} }}$ on
${\mathbb{R}}^{+}\times{\mathbb{R}}^{+}$ is given by
\[
    \Sigma({L^{ \mathtt{3p} }}) = \bigcup\limits_{l\in{\mathbb{N}}_{0}} {\mathcal{C}^{ \mathtt{3p} }_{l}},
    \quad\text{where}\quad
    {\mathcal{C}^{ \mathtt{3p} }_{l}} := \bigcup\limits_{j\in{\mathbb{N}}_{0}}{\mathcal{C}^{ \mathtt{3p\pm} }_{j,l}},\quad l\in{\mathbb{N}}_{0},
\]
and for $k, l \in {\mathbb{N}}_{0}$,
\begin{gather*}
{\mathcal{C}^{ \mathtt{3p\pm} }_{k,l}}  :=
\left\{ (\alpha, \beta) \in {S^{ \mathtt{P\xi} }_{k}}\cap{S^{ \mathtt{DN} }_{l}}:
  \left(F_{k,l}(\alpha, \beta) - \frac{\pi + \eta}{2\pi}\right)
  \left(F_{k,l}(\beta, \alpha) - \frac{\pi + \eta}{2\pi}\right) = 0
  \right\},
\\
  F_{k,l}(\alpha, \beta)  :=
  \frac{\xi}{\pi}\frac{\sqrt{\beta}}{\sqrt{\alpha}+\sqrt{\beta}} +
  \frac{l-k}{2\sqrt{\alpha}} + \frac{l+k+1}{2\sqrt{\beta}}\,.
\end{gather*}

\label{propozice_3p}
\end{proposition}

For the proof of the above proposition see \cite{holnec}.


Let us close this section by summing up the main properties of the {Fu\v{c}\'{\i}k} spectrum
$\Sigma({L})$ which make the operator ${L}$ unique and interesting
(see Figure \ref{obr_FSP_fourp}):
\begin{enumerate}
\item monotonicity and smoothness of the {Fu\v{c}\'{\i}k}~branches are not preserved;
\item the intersection points of $\Sigma({L^{ \mathtt{P}{\xi} }})$ and $\Sigma({L^{ \mathtt{DN} }})$ are bifurcation points
      of new fragments which belong to $\Sigma({L^{ \mathtt{3p} }}) \subset \Sigma({L})$;
\item the intersection points of $\Sigma({L^{ \mathtt{P}{\eta} }})$ and $\Sigma({L^{ \mathtt{DN} }})$ are of no importance;
\item interesting patterns of $\Sigma({L})$ can be observed for different settings of $\xi$
      and $\eta$, the {Fu\v{c}\'{\i}k}~branches intersect away from the diagonal and if we
      continuously change $(\alpha, \beta) \in \Sigma({L})$, the nodal
      properties of the corresponding {Fu\v{c}\'{\i}k}~eigenfunctions are not preserved.
\end{enumerate}

\section{The {Fu\v{c}\'{\i}k} spectrum of the adjoint operator}
\label{sekce4}

In this section, let us focus on the {Fu\v{c}\'{\i}k} spectrum of the
adjoint operator of ${L}$ from the previous section.
Using Theorem 1 in \cite{locker}, we find that
the adjoint operator ${L^{*}}: {\mathrm{dom}}({L^{*}}) \subset L^{2}(0, \pi) \to L^{2}(0, \pi)$ of ${L}$ is
the multi-point differential operator given by
\[
{L^{*}} u(x) = -u''(x),\quad
{\mathrm{dom}}({L^{*}}) =
\{ u \in H^{2}({\mathcal{P}}): B_{i}^{*}(u) = 0,\ i = 1,\dots,6 \},
\]
where ${\mathcal{P}} = \{0, \eta, \xi, \pi\}$ is the partition of the interval $[0, \pi]$,
$u = (u_{1}, u_{2}, u_{3})$, and the adjoint multi-point boundary values are given by
\begin{gather*}
B_{1}^{*}(u) = u_{2} (\xi)  - u_{3} (\xi) + u_{1}(0), \quad
B_{2}^{*}(u) = u_{1} (\eta) - u_{2} (\eta), \\
B_{3}^{*}(u) = u_{2}'(\xi)  - u_{3}'(\xi), \quad
B_{4}^{*}(u) = u_{1}'(\eta) - u_{2}'(\eta) + u_{3}'(\pi),\\
B_{5}^{*}(u) = u_{1}'(0),\quad
B_{6}^{*}(u) = u_{3}(\pi).
\end{gather*}
The {Fu\v{c}\'{\i}k} spectrum problem for the adjoint operator
${L^{*}} u = \alpha u^{+} - \beta u^{-}$ can be written in the form of
the following four-point boundary-value problem
\begin{equation}
\begin{gathered}
  u''(x) + \alpha u^{+}(x) - \beta u^{-}(x) = 0, \quad
  x\in (0, \pi)\setminus \{\xi,\eta\},\\
  u'(0) = u(\pi) = 0,\\
  u(\eta-) = u(\eta+),\quad u'(\xi-) = u'(\xi+),\\
  u(\xi-) = u(\xi+) + u(0),\quad
u'(\eta+) = u'(\eta-) + u'(\pi), \quad \xi\in (0, \pi),\;
 \eta\in (0, \pi).
\end{gathered}
\label{sekce4_mbvp}
\end{equation}


\begin{figure}[ht]
\begin{center}
  \setlength{\unitlength}{0.78mm}
  \begin{picture}(152.7, 50)(0, 0)
    \put(  0, 0){\includegraphics[width=3.9cm]{fig4a}} %{HOL_NEC_fig_4a.eps}}
    \put( 52, 0){\includegraphics[width=3.9cm]{fig4b}} %{HOL_NEC_fig_4b.eps}}
    \put(104, 0){\includegraphics[width=3.9cm]{fig4c}} %{HOL_NEC_fig_4c.eps}}
    \put(140,-3){\makebox(0,0)[cm]{\scriptsize$\sqrt{\alpha}$}}
    \put( -4,40){\makebox(0,0)[cm]{\scriptsize$\sqrt{\beta}$}}
  \end{picture}
\end{center}
\caption{The {Fu\v{c}\'{\i}k} spectrum $\Sigma({L^{*}})$ for three different symmetrical settings of $\xi$ and $\eta$
         with respect to the midpoint of $[0, \pi]$.}
\label{sekce4_obr_1}
\end{figure}


\begin{figure}[ht]
\begin{center}
  \setlength{\unitlength}{0.78mm}
  \begin{picture}(152.7, 50)(0, 0)
    \put(  0, 0){\includegraphics[width=3.9cm]{fig5a}} %{HOL_NEC_fig_5a.eps}}
    \put( 52, 0){\includegraphics[width=3.9cm]{fig5b}} %{HOL_NEC_fig_5b.eps}}
    \put(104, 0){\includegraphics[width=3.9cm]{fig5c}} %{HOL_NEC_fig_5c.eps}}
    \put(140,-3){\makebox(0,0)[cm]{\scriptsize$\sqrt{\alpha}$}}
    \put( -4,40){\makebox(0,0)[cm]{\scriptsize$\sqrt{\beta}$}}
  \end{picture}
\end{center}
\caption{The {Fu\v{c}\'{\i}k} spectrum $\Sigma({L^{*}})$ for $\xi = \pi - \eta$ (left, middle) and for $\xi \neq \pi - \eta$ (right).}
\label{sekce4_obr_2}
\end{figure}

\begin{figure}[ht]
\begin{center}
  \setlength{\unitlength}{0.78mm}
  \begin{picture}(152.7, 50)(0, 0)
    \put(  0, 0){\includegraphics[width=3.9cm]{fig6a}} %{HOL_NEC_fig_6a.eps}}
    \put( 52, 0){\includegraphics[width=3.9cm]{fig6b}} %{HOL_NEC_fig_6b.eps}}
    \put(104, 0){\includegraphics[width=3.9cm]{fig6c}} %{HOL_NEC_fig_6c.eps}}
    \put(140,-3){\makebox(0,0)[cm]{\scriptsize$\sqrt{\alpha}$}}
    \put( -4,40){\makebox(0,0)[cm]{\scriptsize$\sqrt{\beta}$}}
  \end{picture}
\end{center}
\caption{The {Fu\v{c}\'{\i}k} spectrum $\Sigma({L})$ (left) and $\Sigma({L^{*}})$ (middle) and their overlapping (right).}
\label{sekce4_obr_3}
\end{figure}

\begin{figure}[ht]
\begin{center}
  \setlength{\unitlength}{0.78mm}
  \begin{picture}(152.7, 50)(0, 0)
    \put(  0, 0){\includegraphics[width=3.9cm]{fig7a}} %{HOL_NEC_fig_7a.eps}}
    \put( 52, 0){\includegraphics[width=3.9cm]{fig7b}} %{HOL_NEC_fig_7b.eps}}
    \put(104, 0){\includegraphics[width=3.9cm]{fig7c}} %{HOL_NEC_fig_7c.eps}}
    \put(140,-3){\makebox(0,0)[cm]{\scriptsize$\sqrt{\alpha}$}}
    \put( -4,40){\makebox(0,0)[cm]{\scriptsize$\sqrt{\beta}$}}
  \end{picture}
\end{center}
\caption{The overlapping of the {Fu\v{c}\'{\i}k} spectrum $\Sigma({L})$ (orange and black curves) and
         $\Sigma({L^{*}})$ (blue and black curves).}
\label{sekce4_obr_4}
\end{figure}

It is straightforward to verify that the spectrum $\sigma({L^{*}})$ of the adjoint operator ${L^{*}}$
is a countable real discrete spectrum and that
\[
\sigma({L^{*}}) = \sigma({L}) =
\sigma({L^{ \mathtt{P}{\xi} }}) \cup \sigma({L^{ \mathtt{P}{\eta} }}) \cup \sigma({L^{ \mathtt{DN} }}).
\]
Moreover, we have that $\Sigma({L^{ \mathtt{P}{\xi} }}) \cup \Sigma({L^{ \mathtt{P}{\eta} }}) \subset \Sigma({L^{*}})$.
But, the complete explicit analytical description of the {Fu\v{c}\'{\i}k} spectrum $\Sigma({L^{*}})$
seems to be still an open problem. On the other hand, we can use the numerical
continuation techniques combined with the shooting method in order to explore the structure
of the {Fu\v{c}\'{\i}k} spectrum $\Sigma({L^{*}})$ and to formulate new conjectures.
Thus, let us consider the following three initial value problems which corresponds to
the problem \eqref{sekce4_mbvp}
\[
\left\{
\begin{array}{ll}
  u_{1}'' + \alpha u_{1}^{+} - \beta u_{1}^{-} = 0,\\
  u_{1}(0) = A,\\
  u_{1}'(0) = 0,
\end{array}
\right.
\;
\left\{
\begin{array}{ll}
  u_{2}'' + \alpha u_{2}^{+} - \beta u_{2}^{-} = 0,\\
  u_{2}(\eta) = u_{1}(\eta),\\
  u_{2}'(\eta) = u_{1}'(\eta) + B,
\end{array}
\right.
\;
\left\{
\begin{array}{ll}
  u_{3}'' + \alpha u_{3}^{+} - \beta u_{3}^{-} = 0,\\
  u_{3}(\xi) = u_{2}(\xi) + A,\\
  u_{3}'(\xi) = u_{2}'(\xi),
\end{array}
\right.
\]
where $A, B \in {\mathbb{R}}$.
To obtain a point $(\alpha, \beta) \in \Sigma({L^{*}})$, it is enough to find
$(\alpha, \beta, A, B)\in{\mathbb{R}}^{4}$ such that $u_{3}(\pi) = 0$ and $u_{3}'(\pi) = B$.
Since ${L^{*}}$ is the positively homogeneous operator, we can restrict the values of $B$
without any loss of generality to be equal to $-1$, $0$ and $1$.
Figure \ref{sekce4_obr_1} illustrates the numerical approximation of three nontrivial patterns of
the {Fu\v{c}\'{\i}k} spectrum $\Sigma({L^{*}})$ for $\xi = \pi - \eta$ and $\eta = \eta_{1}, \eta_{2}, \eta_{3}$
with $\eta_{1} > \eta_{2} > \eta_{3}$, which enables us to formulate the following conjecture
(see Figure \ref{sekce4_obr_1}, middle).

\begin{conjecture}
For all $i\in{\mathbb{N}}$ and all $j\in{\mathbb{N}}_{0}$ there exist $\xi \in (0, \pi)$ and $\eta = \pi - \xi$ such that
\[
  {\lambda^{ \mathtt{P}{\xi} }_{j}} < {\lambda^{ \mathtt{DN} }_{i}} < {\lambda^{ \mathtt{P}{\xi} }_{j+1}},\quad
  \left\{
  ({\lambda^{ \mathtt{P}{\xi} }_{j}}, {\lambda^{ \mathtt{P}{\xi} }_{j}}), ({\lambda^{ \mathtt{DN} }_{i}}, {\lambda^{ \mathtt{DN} }_{i}}), ({\lambda^{ \mathtt{P}{\xi} }_{j+1}}, {\lambda^{ \mathtt{P}{\xi} }_{j+1}})
  \right\} \subset C,
\]
where $C$ is the component of $\Sigma({L^{*}})$.
\end{conjecture}

Figure \ref{sekce4_obr_2} illustrates the pathological patterns of
the {Fu\v{c}\'{\i}k} spectrum $\Sigma({L^{*}})$ with ``nonstandard'' behavior:
the intersection of the {Fu\v{c}\'{\i}k}~branches away from the diagonal,
monotonicity of the {Fu\v{c}\'{\i}k}~branches is not preserved;
some fragments of $\Sigma({L^{*}})$ connect
the consecutive periodic {Fu\v{c}\'{\i}k}~curves of $\Sigma({L^{ \mathtt{P}{\xi} }})\cup\Sigma({L^{ \mathtt{P}{\eta} }})$.

Now, if we compare the numerical approximation of the {Fu\v{c}\'{\i}k} spectrum $\Sigma({L^{*}})$ to
the {Fu\v{c}\'{\i}k} spectrum $\Sigma({L})$ given explicitly by \eqref{sekce3_eq_1} and Proposition \ref{sekce3_prop_1},
we can formulate the following conjecture (see Figures \ref{sekce4_obr_3} and \ref{sekce4_obr_4}).

\begin{conjecture}$\,$
\begin{enumerate}
  \item $\Sigma({L^{*}}) \neq \Sigma({L})$ inspite of the fact that $\sigma({L^{*}}) = \sigma({L})$;
  \item there exist pairs $(\alpha, \beta) \in \Sigma({L}) \cap \Sigma({L^{*}})$
        with $\alpha \neq \beta$ such that $(\alpha, \beta)\not\in\Sigma({L^{ \mathtt{P}{\xi} }}) \cup \Sigma({L^{ \mathtt{P}{\eta} }})$.
\end{enumerate}
\end{conjecture}

Let us note that Figure \ref{sekce4_obr_4} is the combination of
Figures \ref{obr_FSP_fourp} and \ref{sekce4_obr_2}
and illustrates which parts of the {Fu\v{c}\'{\i}k}
spectra $\Sigma({L})$ and $\Sigma({L^{*}})$
coincide.

\subsection*{Acknowledgements}
The authors were supported by the Ministry of Education of the Czech Republic, Research Plan \# MSM4977751301
and partially by grant no. ME09109, Program KONTAKT.

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\end{document}