\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 156, pp. 1--39.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/156\hfil Nodal solutions]
{Nodal solutions for singular second-order boundary-value problems}

\author[A. Benmeza\"i, W. Esserhane, J. Henderson \hfil EJDE-2014/156\hfilneg]
{Abdelhamid Benmeza\"i, Wassila Esserhane, Johnny Henderson}  % in alphabetical order

\address{Abdelhamid Benmeza\"i \newline
Faculty of Mathematics, USTHB, Algiers, Algeria}
\email{aehbenmezai@gmail.com}

\address{Wassila Esserhane \newline
Graduate School of Statistics and Applied Economics,
P.O. Box 11, Doudou Mokhtar, Ben-Aknoun Algiers, Algeria}
\email{ewassila@gmail.com}

\address{Johnny Henderson \newline
Department of Mathematics, Baylor University, Waco,
Texas 76798-7328, USA}
\email{Johnny\_Henderson@baylor.edu}

\thanks{Submitted January 27, 2014. Published July 7, 2014.}
\subjclass[2000]{34B15, 34B16, 34B18}
\keywords{Singular second-order BVPs; nodal solutions;
\hfill\break\indent global bifurcation theorem}

\begin{abstract}
 We use a global bifurcation theorem to prove the existence of
 nodal solutions to the singular second-order two-point boundary-value
 problem
 \begin{gather*}
 -( pu') '(t)=f(t,u(t))\quad t\in ( \xi ,\eta) , \\
 au(\xi )-b\lim_{t\to\xi} p(t)u'(t)=0, \\
 cu(\eta )+d\lim_{t\to\eta} p(t)u'(t)=0,
 \end{gather*}
 where $\xi ,\eta $, $a,b,c,d$ are real numbers with $\xi <\eta$,
 $a,b,c,d\geq 0$ , $p:( \xi ,\eta ) \to [ 0,+\infty) $ is a measurable
 function with $\int_{\xi }^{\eta }1/p(s)\,ds<\infty $ and
 $f:[ \xi ,\eta ] \times [ 0,+\infty) \to [ 0,+\infty ) $ is a Carath\'eodory
 function.
\end{abstract}

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


\section{Introduction}

Many articles concerning the existence of nodal solutions for second-order
differential equations subject to various boundary conditions, have appeared
during the previous five decades; see for example
\cite{Ber1,Ber2,cui,genoud,rynne1,rynne5,ma5,ma3,ma1,rma,ma4,ma2,nait,
rab11,rab12,rab13,rynne2,rynne20,rynne3,rynne5.1,rynne6} and references therein.

Ma and Thompson \cite{rma,ma4,ma2} considered the
boundary-value problem (bvp for short),
\begin{equation}
\begin{gathered}
-u''=a( t) f(u),\quad t\in  ( 0,1) , \\
u(0)=u(1)=0
\end{gathered}  \label{000}
\end{equation}
where $a:[ 0,1] \to [ 0,+\infty ) $ is
continuous and does not vanish identically, and $f:\mathbb{R}\to\mathbb{R}$
is continuous with $f( s) s>0$ for $s\neq 0$. They proved also,
that bvp \eqref{000} admits $2k$ nodal solutions when the interval whose
extremities are $\lim_{u\to 0}f(u)/u$ and
$\lim_{|u| \to +\infty }f(u)/u$ contains $k$ eigenvalues
of the linear bvp associated with \eqref{000},
\begin{gather*}
-u''=\lambda a( t) u,\quad t\in ( 0,1) ,\\
u(0)=u(1)=0.
\end{gather*}

Articles \cite{ma3} and \cite{rynne3} were devoted to the multipoint bvp,
\begin{equation}
\begin{gathered}
-u''=f(u),\quad t\in ( 0,1) , \\
u(0)=0,\ u(1)=\sum_{i=1}^{m-2}  \alpha _iu( \eta _i)
\end{gathered}  \label{bvp01}
\end{equation}
where $f:\mathbb{R}\to\mathbb{R}$ is $C^{1}$ with
$f( 0) =0$, $m\geq 3$, $\eta _i\in (0,1)$ and $\alpha _i>0$
 for $i=1,\dots ,m-2$ with $\sum_{i=1}^{m-2} \alpha _i<1$,
by which Rynne \cite{rynne3} extended
the result and filled some gaps in \cite{ma3}.

Roughly speaking, Rynne proved that bvp \eqref{bvp01} admits $2k$ nodal
solutions when the interval whose extremities are
$\lim_{u\to 0} f(u)/u$ and $\lim_{| u| \to +\infty }
f(u)/u$ contains $k$ eigenvalues of the linear bvp associated with
\eqref{bvp01},
\begin{gather*}
-u''=\lambda u,\quad t\in ( 0,1) , \\
u(0)=0,\quad u(1)=\sum_{i=1}^{m-2}  \alpha _iu( \eta_i) .
\end{gather*}

This result was extended by Genoud and Rynne in \cite{rynne5} to the case
with variable coefficients.

Existence and multiplicity of positive solutions for second order bvps
having singular dependence on the independent variable, have been considered
in many papers; see, for example,
\cite{ntouyas,BGK,sb100,sb2,sb4,sb5,bib51,bib50,bib52, sb7}
and references therein. In particular, it is proved in
\cite{sb2,bib51,bib50} that, if the function $a$ in bvp
\eqref{000} is just continuous on $( 0,1) $ and satisfies
\begin{equation}
\int_0^{1}t( 1-t) a(t)dt<\infty ,  \label{01}
\end{equation}
then  \eqref{000} admits one or more positive solutions under some
additional conditions on the behavior of the ratio $f(u)/u$ at $0$ and
$+\infty $.
A natural question becomes,
\begin{quote}
Is it possible to obtain
existence results for nodal solutions to bvp \eqref{000} under Hypothesis
\eqref{01}?
\end{quote}
So, the main goal of this paper is to give an answer to this question.

In fact, we will give an answer  for a
more general bvp having a nonlinearity  more general than \eqref{000}, under
a hypothesis looking like \eqref{01}. This answer will be based on the
knowledge of the spectrum of the linear problem associated with the
nonlinear bvp. This was the case also for all the works in
\cite{Ber1,rynne1,rynne5,ma5,ma3,ma1,rma,ma4,ma2,rynne2,rynne3,rynne5.1,rynne6}.

We need also in this work to introduce the concept of half-eigenvalue which
generalizes the notion of eigenvalue. The definition of half-eigenvalue here
is not the same given by Berysticki (see Remark \ref{XXL}), and for the role
that will be played by this notion, we refer the reader to
\cite{Ber1,binding,rynne1,rynne20,rynne3,rynne5.1}.

A typical example of a weight function satisfying Hypothesis \eqref{01} is
$a( t) =t^{-3/2}( 1-t) ^{-3/2}$.
Note that such a weight $a$ is not integrable near $0$ and $1$.
We have a similar situation
in this work and this causes many difficulties in proving existence of
half-eigenvalues as well as in proving the main results of this paper. The
existence of half-eigenvalues will be obtained by sequential arguments.
We will use in this work, the global bifurcation theorem of Rabinowitz to
obtain our main results.

\section{Main results}

This article concerns the existence of nodal solutions for the bvp,
\begin{equation}
\begin{gathered}
-( pu') '(t)=f(t,u(t)),\ \text{a.e.}\ t\in
( \xi ,\eta ) \\
au(\xi )-b\lim_{t\to \xi } p(t)u'(t)=0, \\
cu(\eta )+d\lim_{t\to \eta} p(t)u'(t)=0,
\end{gathered}  \label{bvp1}
\end{equation}
where $\xi ,\eta \in\mathbb{R}$ with $\xi <\eta$,
$a,b,c,d\in\mathbb{R} ^{+}=[ 0,+\infty ) $,
$p:( \xi ,\eta ) \to\mathbb{R}^{+}$ is a measurable function and
$f:( \xi ,\eta )\times \mathbb{R}\to \mathbb{R}$ is a Carath\'{e}odory function
($f(\cdot ,u)$ is measurable for $u$ fixed and $f(t,\cdot )$ is continuous for
$t\in ( \xi ,\eta ) $ a.e.).

Throughout this article, we assume that
\begin{gather}
\int_{\xi }^{\eta }\frac{d\tau }{p(\tau )}<\infty ,  \label{hyp1}\\
\Delta =ad+ac\int_{\xi }^{\eta }\frac{d\tau }{p(\tau )}+bc>0.  \label{hyp2}
\end{gather}
Let
\[
L_{G}^{1}[ \xi ,\eta ] =\big\{ q:( \xi ,\eta )
\to \mathbb{R} \text{ measurable, } \int_{\xi }^{\eta }G(
t,t) | q(t)| dt<\infty \big\}
\]
and let $ K_{G}$ be the cone of all functions $q\in L_{G}^{1}[ \xi ,\eta ] $
such that $ q(t)\geq 0$ a.e. $ t\in [ \xi ,\eta ] $ and
 $ q>0$ in a subset of a positive measure of $[ \xi ,\eta ] $ where
\[
G(t,s)=\frac{1}{\Delta }
\begin{cases}
\Phi _{ab}(s)\Psi _{cd}(t), & \xi \leq s\leq t\leq \eta , \\
\Phi _{ab}(t)\Psi _{cd}(s), & \xi \leq t\leq s\leq \eta .
\end{cases}
\]
is the Green's function associated with the bvp
\begin{gather*}
-( pu') '(t)=0,\,\text{a.e.}\,t\in ( \xi,\eta ) , \\
au(\xi )-b\,\lim_{t\to \xi }p(t)u'(t)=0, \\
cu(\eta )+d\,\lim_{t\to \eta }p(t)u'(t)=0,
\end{gather*}
and the functions
$\Phi _{ab}(t)=b+a\int_{\xi }^{t} 1/p(\tau )\, d\tau$
and $\Psi _{cd}(t)=d+c\int_{t}^{\eta } 1/p(\tau )\, d\tau$
are well defined on $[\xi ,\eta ]$.

Note that the space $L_{G}^{1}$  $[ \xi ,\eta ] $
depends on the parameters $b$ and $d$.
In fact, we have that
$L_{G}^{1}[ \xi ,\eta ]=L^{1}[ \xi,\eta ] $ if
$bd\neq 0$ and $L_{G}^{1}[ \xi ,\eta] \backslash L^{1}[ \xi ,\eta ] $ is
nonempty if $bd=0$. More precisely, we have that $q\in L_{G}^{1}$
 $[ \xi ,\eta ] $ is not integrable at $\xi $ if and
only if $b=0$ and $q$ is not integrable at $\eta $ if and
only if $d=0$. For example, if $p=1$ and $b=d=0$ the function
$q( s) =( s( 1-s) ) ^{-3/2}\in L_{G}^{1}[ \xi ,\eta ] \backslash L^{1}
[ \xi ,\eta ]$.
Moreover, we have that
$L_{G}^{1}[ \xi ,\eta ] \subset L_{\rm loc}^{1}( \xi,\eta) $.

The main result of this article (Theorem \ref{thm3}) will be obtained under
the following additional conditions on the nonlinearity $f$:

There exist functions
$\alpha _{\infty },\beta _{\infty },\gamma _{\infty},\delta _{\infty }$ and
$q_0$ in $K_{G}$ such that the set
\[
\{ t\in ( \xi ,\eta ) :\alpha _{\infty }( t) \beta _{\infty }( t) >0\}
\]
is of a positive measure,
\begin{equation}
\begin{gathered}
\lim_{u\to 0}\frac{f(t,u)}{u}=q_0( t) \quad \text{for }
t\in [ \xi ,\eta ] \ a.e., \\
\lim_{u\to -\infty }\frac{f(t,u)}{u}=\beta _{\infty }(t)\quad \text{for }
t\in [ \xi ,\eta ] \ a.e., \\
\lim_{u\to +\infty }\frac{f(t,u)}{u}=\alpha _{\infty }(t)\quad
\text{for } t\in [ \xi ,\eta ] \text{ a.e. }
\end{gathered}
\label{hyp3}
\end{equation}
and
\begin{equation}
\delta _{\infty }( t) \leq \frac{f( t,u) }{u}\leq
\gamma _{\infty }( t) \quad \text{for all }u\in \mathbb{R}
\text{ and }t\in [ \xi ,\eta ] \text{ a.e. }
\label{hyp4}
\end{equation}

From all the above hypotheses, we understand that a solution to bvp
\eqref{bvp1} is a function
$u\in C[ \xi ,\eta ] \cap C^{1}( \xi,\eta ) $ with
$( pu') '\in L_{G}^{1} [ \xi ,\eta ] $, satisfying all equations in \eqref{bvp1}.

\begin{remark} \label{rmk2.1} \rm
Note that Hypothesis \eqref{hyp4} implies that the nonlinearity $f$
satisfies the following sign condition:
\[
f( t,u) u\geq 0\quad \text{for all $u\in\mathbb{R}$ and
$t\in [ \xi ,\eta ]$ a.e.}
\]
\end{remark}

\begin{example}\label{examp1} \rm
A typical example of a nonlinearity satisfying Hypotheses
\eqref{hyp3})-\eqref{hyp4}, when $p=1$ and $b=d=0$, is
\begin{align*}
f( t,u) &=At^{-3/2}( 1-t) ^{-5/4}u+Bt^{-7/6}
( 1-t) ^{-7/4}\frac{u^{3}}{1+u^{2}+e^{-u}} \\
&\quad +Ct^{-11/7}( 1-t) ^{-13/10} \frac{u^{3}}{1+u^{2}+e^{u}},
\end{align*}
where $A,B,C$ are positive constants.
\end{example}


Throughout this article, we denote by $E$ the Banach space of all continuous
functions defined on $[ \xi ,\eta ] $, equipped with the sup-norm
denoted $\| \cdot \| _{\infty }$ and by $Y$ the Banach
space defined as
\begin{align*}
Y=\Big\{&v\in AC[ \xi ,\eta ] : pv'\in C[ \xi ,\eta ] \text{ and }  \\
&av(\xi )-b \lim_{t\to \xi }p(t)v'(t)=cv(\eta )+d\lim_{t\to\eta} p(t)v'(t)=0
\Big\}
\end{align*}
equipped with the norm $\| v\| _{Y}=\| v\|_{\infty }+\| pv'\| _{\infty }$ for
$v\in Y$. In all this paper, $\pounds $ is the differential operator
given  by
\[
\pounds u(x)=-(pu')'(x)
\]
with domain
\[
D( \pounds ) =\big\{ v\in AC[ \xi ,\eta ] :pv'\in C( \xi ,\eta )  \text{ and }
( pv')'\in L_{G}^{1}[ \xi ,\eta ] \big\} .
\]
Set
\[
Y_{\#}=\big\{ v\in D( \pounds ) :av(\xi )-b\lim_{t\to \xi }p(t)v'(t)=cv(\eta )+d\ \underset{
t\to \eta }{\lim }\ p(t)v'(t)=0\big\} .
\]
We have that $\pounds :Y_{\#}\to L_{G}^{1}[ \xi ,\eta ] $
is one to one, with
\[
\pounds ^{-1}v( t) =\int_{\xi }^{\eta }G( t,s) v(
s) ds\text{ for all }v\in L_{G}^{1}[ \xi ,\eta ] .
\]

For $u\in AC[ \xi ,\eta ] $, $u^{[ 1] }$ is the
quasiderivative of $u$, for $t\in [ \xi ,\eta ]$; that is,
$u^{[ 1] }( t) =\lim_{\tau \to t}p(\tau )u'(\tau )$ when it exists.

For $k\geq 1$, let $S_k^{+}$ denote the set of all functions
$v\in AC[\xi ,\eta ] $ with $pv'\in C( \xi ,\eta )$,
having exactly $(k-1)$ simple zeros in $( \xi ,\eta ) $
(if $v(\tau )=0$ then $v^{[ 1] }(\tau )\neq 0$) and $v$ is positive in
a right neighborhood of $\xi $, and denote $S_k^{-}=-S_k^{+}$ and
$S_k=S_k^{+}\cup S_k^{-}$.

Let
\begin{gather*}
\rho _0=\Big( \int_{\xi }^{\eta }\frac{ds}{p(s)}\Big) ^{-1}( \eta
-\xi ) , \\
C_{\#}^{1}[ \xi ,\eta ] =\big\{ v\in C^{1}[ \xi ,\eta
] :av(\xi )-b\rho _0\,v'(\xi )=0 \text{ and }
 cv(\eta)+d\,\rho _0v'(\eta )=0\big\}
\end{gather*}
equipped with the $C^{1}$-norm and, for all $k\geq 1$ let $\Theta _k^{+}$
be the set of all functions $v\in C_{\#}^{1}[ \xi ,\eta ] $
having exactly $(k-1)$ simple zeros in $( \xi ,\eta ) $ and $v$
is positive in a right neighborhood of
$\xi , \Theta _k^{-}=-\Theta_k^{+}$ and
$\Theta _k=\Theta _k^{+}\cup \Theta _k^{-}$. It is well
known that $\Theta _k^{+},\Theta _k^{-}$ and $\Theta _k$ are open sets
in $C_{\#}^{1}[ \xi ,\eta ] $. Since for all $k\geq 1$ and
$\nu=+ $ or $-$, $\Phi ( S_k^{\nu }\cap Y) =\Theta _k^{\nu }$
where $\Phi :Y\to C_{\#}^{1}[ \xi ,\eta ] $ is the
homeomorphism between Banach spaces defined by
\[
\Phi ( u) =u\circ \varphi ^{-1}\quad \text{with }
 \varphi (t) =\xi +\rho _0\int_{\xi }^{t}\frac{ds}{p(s)},
\]
we have that $S_k^{\nu }\cap Y$ is an open set in $Y$.
 Moreover, since if $u\in \partial \Theta _k^{\nu }$ then there exists
$\tau \in [ \xi,\eta ] $ such that $u(\tau )=u'(\tau )=0$, we have that for
all $v\in \partial ( S_k^{\nu }\cap Y) $ there exists
$\tau \in [ \xi ,\eta ] $ such that
$u(\tau )=u^{[ 1] }(\tau )=0$.

For $\nu =+$ or $-$, let $I^{\nu }:E\to E$ be defined
by $\ I^{\nu }u( x) =\max ( \nu u(x),0)$, for $u\in E$.
For all $u\in E$, we have
\[
u=I^{+}u-I^{-}u\quad \text{and}\quad | u| =I^{+}u+I^{-}u.
\]
This implies that, for all $u,v\in E$,
\begin{equation} \label{ttt}
\begin{gathered}
| I^{+}u-I^{+}v| \leq \frac{| u-v| }{2}+\frac{| | u| -| v|| }{2}\leq | u-v| \,, \\
| I^{-}u-I^{-}v| \leq \frac{| u-v| }{2}+\frac{| | u| -| v|| }{2}\leq | u-v| ,
\end{gathered}
\end{equation}
and the operators $I^{+},I^{-}$ are continuous.

For sake of simplicity, throughout this paper,  we will use
$u^{+}$ and $u^{-}$ instead of $I^{+}u$ and $I^{-}u$, respectivley.
Now we focus our attention on the eigenvalue bvp
\begin{equation}
\begin{gathered}
-( pu') '(t)=\lambda ( \alpha (
t) u^{+}(t)-\beta ( t) u^{-}(t)) ,\ \text{a.e.}\ t\in
( \xi ,\eta ) , \\
au(\xi )-b\lim_{t\to\xi} p(t)u'(t)=0, \\
cu(\eta )+d\lim_{t\to\eta} p(t)u'(t)=0,
\end{gathered}  \label{bvp2}
\end{equation}
where $\alpha $ and $\beta $ are functions in $K_{G}$ such that the set
$\{t\in ( \xi ,\eta ) :\alpha (t) \beta ( t) >0\}$
is of a positive measure, and $\lambda $ is a real parameter.

\begin{definition} \label{def2.3} \rm
We say that $\lambda $ is a half-eigenvalue of \eqref{bvp2} if there exists
a nontrivial solution $( \lambda ,u_{\lambda }) $ of \eqref{bvp2}.
In this situation, $\{ (\lambda ,tu_{\lambda }),\text{ }t>0\} $
is a half-line of nontrivial solutions of \eqref{bvp2} and $\lambda $ is
said to be simple if all solutions $(\lambda ,v)$ of \eqref{bvp2} with $v$
and $u$ having the same sign on a right neighborhood of $\xi $ are on this
half-line. There may exist another half-line of solutions
$\{ (\lambda,tv_{\lambda }),\;t>0\} $, but then we say that $\lambda $ is
simple if $u_{\lambda }$ and $v_{\lambda }$ have different signs on a right
neighborhood of $\xi $ and all solutions $(\lambda ,v) $ of \eqref{bvp2}
lie on these two half lines.
\end{definition}

\begin{theorem}\label{thm10}
Assume that \eqref{hyp1} and \eqref{hyp2} hold, and
$\alpha ,\beta \in K_{G}\cap L^{1}[ \xi ,\eta ]$. Then the set of
half-eigenvalues to bvp \eqref{bvp2} consists of two increasing sequences
$(\lambda _k^{+})_{k\geq 1}$ and $(\lambda _k^{-})_{k\geq 1}$ such that
for all $k\geq 1$ and $\nu =+$ or $-$,
\begin{enumerate}
\item $\lambda _k^{\nu }$ is simple and is the unique half-eigenvalue
having a half-line of solutions in
$\{ \lambda _k^{\nu }\}\times S_k^{\nu }$.

\item $\lambda _k^{\nu }$ is a nondecreasing function with the respect of
each of the weights $\alpha $ and $\beta $ lying in $L^{1}[ \xi ,\eta] $.
\end{enumerate}
\end{theorem}

\begin{theorem}\label{thm1}
Assume that \eqref{hyp1} and \eqref{hyp2} hold. Then the set of
half-eigenvalues to bvp \eqref{bvp2} consists of two nondecreasing sequences
$(\lambda _k^{+})_{k\geq 1}$ and $(\lambda _k^{-})_{k\geq 1}$ such that
for all $k\geq 1$ and $\nu =+$ or $-$, $\lambda _k^{\nu }$ is the unique
half eigenvalue having a half-line of solutions in
$\{ \lambda _k^{\nu }\} \times S_k^{\nu }$. Moreover, for all $k\geq 1$ and
$\nu =+$ or $-$, $\lambda _k^{\nu }$ is a nondecreasing function with the
respect of each of the weights $\alpha $ and $\beta $ lying in
$L_{G}^{1}[ \xi ,\eta ] $.
\end{theorem}

\begin{remark}\label{rema1} \rm
It is clear that for all $k\geq 1$ and $\nu =+$ or $-$,
$\lambda _k^{\nu }$ depends on the weights $p, \alpha , \beta $ and on
$( \xi ,\eta ,a,b,c,d) $. When there is no confusion, we just
denote $\lambda _k^{\nu}$, and when we need to be more precise, we write
$\lambda _k^{\nu }( \alpha ,\beta ) $.
\end{remark}

Consider the bvp,
\begin{equation}
\begin{gathered}
\pounds u(t)=\mu q(t)u(t),\quad \text{a.e. } t\in ( \xi ,\eta ), \\
au(\xi )-b \lim_{t\to \xi }p( t) u'(t)=0, \\
cu(\eta )+d \lim_{t\to \eta }p( t) u'(t)=0,
\end{gathered}  \label{bvp3}
\end{equation}
where $\mu $ is a real parameter, and $q\in K_{G}$.

It is clear that if $\mu $ is an eigenvalue for \eqref{bvp3} then $\mu $
depends on the weights $p,q$ and on $( \xi ,\eta ,a,b,c,d) $.
When there is no confusion, we just denote $\mu ( q)$, and when
we need to be more precise, we write $\mu ( q,[ \xi ,\eta ]) $.

\begin{theorem} \label{thm2}
Assume that \eqref{hyp1} and \eqref{hyp2} hold. Then bvp \eqref{bvp3}
admits a sequence of eigenvalues $(\mu _k(q))_{k\geq 1}$ such that:
\begin{enumerate}
\item For all $k\geq 3$, $\mu _k(q)$ is simple and the associated
eigenfunction $\phi _k\in S_k$.

\item For all $k\geq 3$, $\mu _k(q)<\mu _{k+1}(q)$.

\item If $bd\neq 0$, or $bd=0$ and $q\in L^{1}[ \xi ,\eta ] $,
then $\mu _1(q,[ \xi ,\eta ] )<\mu _2(q,[ \xi ,\eta
] )$ and $\mu _1(q),\mu _2(q)$ are simple having eigenvectors
respectively in $S_1$ and $S_2$. If $bd=0$ and
$q\notin L^{1}[ \xi ,\eta ] $, then $\mu _1(q)=\mu _2(q)$ and
$\mu _1(q)=\mu _2(q)$
is double having two eigenvectors $\phi _{1,1}\in S_1$ and
$\phi _{1,2}\in S_2$.

\item For all $k\geq 1$ and $\theta >0$, $\mu _k(\theta q)=\frac{\mu
_k(q)}{\theta }$.

\item Let $q_1\in K_{G}$. We have $\mu _k(q_1)\geq \mu _k(q)$ for
all $k\geq 1$ whenever $q_1\leq q$.

\item If $[ \xi _1,\eta _2] \subset [ \xi ,\eta ] $
then $\mu _k(q,[ \xi ,\eta ] )\leq \mu _k(q,[ \xi_1,\eta _1] )$.

\item For all $k\geq 1$, $\mu _k(\cdot ,[ \xi ,\eta ]
):K_{G}\to\mathbb{R}$ is continuous.
\end{enumerate}
\end{theorem}

\begin{remark} \label{rmk2.8} \rm
Since the weight $q$ in Theorem \ref{thm2} is not necessarily in
$L^{1}[ \xi ,\eta ] $, Theorem \ref{thm2} is not covered by
\cite[Theorems 4.3.1, 4.3.2, 4.3.3, 4.3.4]{zettl}.
\end{remark}

For the statement of the main results of this paper we introduce the
following notation:
\[
\theta _{\infty }( t) =\max ( \alpha _{\infty }(t) ,\beta _{\infty }( t) ), \quad
\vartheta _{\infty }( t) =\min ( \alpha _{\infty }( t) ,\beta_{\infty }( t) ),
\]
and let $( \mu _k( \theta _{\infty }) ) _{k\geq 1}$,
$( \mu _k( \vartheta _{\infty }) ) _{k\geq 1}$ and $
( \mu _k( q_0) ) _{k\geq 1}$ be respectively the
sequences of eigenvalues given by Theorem \ref{thm2} for
$q=\theta _{\infty}$, $q=\vartheta _{\infty }$ and $q=q_0$.

\begin{theorem} \label{thm3}
Assume that \eqref{hyp1}-\eqref{hyp4} hold and that there exist two
integers $k,l$ with $2<k<l$ such that one of the following situations
holds:
\[
\mu _{l}( q_0) <1<\mu _k( \theta _{\infty })
\]
or
\[
\mu _{l}( \vartheta _{\infty }) <1<\mu _k( q_0)\,.
\]
Then  for all $i\in \{ k,\ldots,l\} $ and $\nu =+$ or $-$,
\eqref{bvp1} has a solution in $S_i^{\nu }$.
\end{theorem}

Consider the separated variables case
\begin{equation}
\begin{gathered}
\pounds u(t)=q_{sv}(t)h( u(t)) ,\quad \text{a.e.}\ t\in (
\xi ,\eta ) \\
au(\xi )-b\lim_{t\to \xi }p( t) u'(t)=0 \\
cu(\eta )+d\lim_{t\to \eta }p( t) u'(t)=0
\end{gathered}  \label{bvp3.1}
\end{equation}
where $q_{sv}$ is a nonnegative function in
$L_{G}^{1}[ \xi ,\eta ] $ which does not vanish identically in $[ \xi ,\eta ] $
and $h:\mathbb{R}\to\mathbb{R}$ is a continuous function such that
\begin{equation}
\begin{gathered}
h( x) x>0\quad \text{for all }x\neq 0 \\lim_{x\to 0}\frac{h( x) }{x}=h_0, \quad
\lim_{x\to +\infty }\frac{h( x) }{x}=h_{+\infty }, \\lim_{x\to -\infty }\frac{h( x) }{x}=h_{-\infty }\quad
\text{with }h_0,h_{+\infty },h_{-\infty }\in ( 0,+\infty ) .
\end{gathered} \label{hyp3.1}
\end{equation}

The following corollary provides an answer to a more general situation than
those studied in \cite{ma4} and \cite{ma2} and also covers
\cite[Theorems 2 and 3]{nait}.

\begin{corollary} \label{coro1}
Assume that \eqref{hyp3.1} holds  and there exist two
integers $k,l$ with $2<k<l$, such that one of the following situations
holds:
\[
h_{+\infty },h_{-\infty }<\mu _k( q_{sv}) <\mu _{l}(
q_{sv}) <h_0
\]
or
\[
h_0<\mu _k( q_{sv}) <\mu _{l}( q_{sv}) <h_{+\infty},h_{-\infty }\,.
\]
Then for all $i\in \{ k,\ldots ,l\} $ and $\nu =+$ or $-$,
 \eqref{bvp3.1} has a solution in $S_i^{\nu }$.
\end{corollary}

\begin{proof}
Set $f(t,u)=q_{sv}(t)h( u) $. It is easy to check that $f$
satisfies Hypotheses \eqref{hyp3}-\eqref{hyp4}, with
\begin{gather*}
\alpha _{\infty }( t) =h_{+\infty }q_{sv}( t) ,\quad
\beta _{\infty }( t) =h_{-\infty }q_{sv}( t), \quad
q_0( t) =h_0q_{sv}( t), \\
\gamma _{\infty }=h_{\sup }q_{sv}( t) ,\quad
\delta _{\infty }=h_{\inf }q_{sv}( t) , \\
h_{\inf } =\inf \{ h(u)/u:u\neq 0\},   \\
h_{\sup } =\sup \{ h(u)/u:u\neq 0\} .
\end{gather*}
Also, we have
\[
\theta _{\infty }( t) =\max ( h_{+\infty },h_{-\infty}) q_{sv}( t),\quad
\vartheta _{\infty }(t) =\min ( h_{+\infty },h_{-\infty }) q_{sv}( t).
\]
Since Property 2 of Theorem \ref{thm2} implies that for all $n\geq 1$,
\[
\mu _n( \theta _{\infty }) =\frac{\mu _n( q_{sv})
}{\max ( h_{+\infty },h_{-\infty }) },\quad
\mu _n( \vartheta_{\infty }) =\frac{\mu _n( q_{sv}) }{\min (
h_{+\infty },h_{-\infty }) }, \quad
\mu _n( q_0) = \frac{\mu _n( q_{sv}) }{h_0},
\]
we obtain that $\mu _{l}( q_0) <1<\mu _k( \theta
_{\infty }) $ if $h_{+\infty },h_{-\infty }<\mu _k(
q_{sv}) <\mu _{l}( q_{sv}) <h_0$ and
$\mu _{l}(\vartheta _{\infty }) <1<\mu _k( q_0) $ if
$h_0<\mu _k( q_{sv}) <\mu _{l}( q_{sv}) <h_{+\infty
},h_{-\infty }$. Thus, the conclusion of Corollary \ref{coro1} follows from
Theorem \ref{thm3}.
\end{proof}

\section{Background}

\subsection{A comparison result}

\begin{theorem}\label{Preli1}
Let $u$ and $v$ be two functions in $S_k^{\nu }$. Then,
there exist two intervals $[\xi _1,\eta _1]$ and $[\xi _2,\eta _2]$
such that $uv\geq 0$ in $[\xi _i,\eta _i]$ $i=1,2$. Moreover if
$pu',pv'\in AC[\xi _i,\eta _i]$ $i=1,2$ then
\[
\int_{\xi _1}^{\eta _1}v\pounds u-u\pounds v\geq 0,\quad
\int_{\xi _2}^{\eta _2}v\pounds u-u\pounds v\leq 0.
\]
\end{theorem}
The proof of this theorem is based on the following lemma.

\begin{lemma}[\cite{Ber1}] \label{Preli2}
Let $j$ and $k$ be two integers such that $j\geq k\geq 2$.
Suppose that there exist two families of real numbers
\begin{gather*}
\xi _0  = \xi <\xi _1<\xi _2<\dots \xi _{k-1}<\xi _k=\eta \\
\eta _0 = \xi <\eta _1<\eta _2<\dots \eta _{j-1}<\eta _{j}=\eta .
\end{gather*}
Then, if $\xi _1\leq \eta _1$ there exist two integers $j_0$ and
$k_0 $ having the same parity, $1\leq j_0\leq j-1$,
$1\leq k_0\leq k-1$ such that
\[
\xi _{k_0}\leq \eta _{j_0}\leq \eta _{j_0+1}\leq \xi _{k_0+1}.
\]
\end{lemma}

\begin{proof}[Proof of Theorem \protect\ref{Preli1}]
The case $k=1$ is obvious. Let
\begin{gather*}
x_0=\xi <x_1<x_2<\dots <x_{k-1}<x_k=\eta,\\
z_0=\xi <z_1<z_2<\dots <z_{k-1}<z_k=\eta \newline
\end{gather*}
be  the sequences of zeros of $u$ and $v$, respectively.

Suppose $x_1\leq z_1$. Then we deduce from Lemma \ref{Preli2} the
existence of integers $k_0$ and $j_0$ having the same parity such that
$x_{k_0}\leq z_{j_0}\leq z_{j_0+1}\leq x_{k_0+1}$.

Therefore, we choose $\xi _1=x_0$, $\eta _1=x_1$, $\xi _2=z_{j0}$
and $\eta _2=z_{j_0+1}$, and since $k_0$ and $j_0$ have the same
parity, $u$ and $v$ have the same sign in both the intervals
$[\xi _1,\eta_1]$ and $[\xi _2,\eta _2]$.

Now if $u,v\in S_k^{\nu }\cap Y$, then for $i=1,2$,
$u^{[ 1]}( \xi _i) $, $v^{[ 1] }( \xi _i)$,
$u^{[ 1] }( \eta _i)$ and $v^{[ 1] }(\eta _i) $ exist and are finite.
Moreover if $u\geq 0$ and $v\geq 0$ in $[ \xi _i,\eta _i] $ $i=1,2$
(the other cases can be checked similarly) then we have since $u$ and $v$
satisfy the boundary condition at $\xi _1$,
\[
v( \xi _1) u^{[ 1] }( \xi _1) -u(
\xi _1) v^{[ 1] }( \xi _1) =0,\ u^{[ 1
] }( \eta _1) \leq 0,\
\]
and
\begin{gather*}
v^{[ 1] }( \xi _2) \geq 0,\quad
 v^{[ 1]}( \eta _2) \leq 0, \\
-v( \eta _2) u^{[ 1] }( \eta _2)
+u( \eta _2) v^{[ 1] }( \eta _2)
=\begin{cases}
0, & \text{if } \eta _2=\eta, \\
u( \eta _2) v^{[ 1] }( \eta _2) \leq
0, & \text{if } \eta _2<\eta .
\end{cases}
\end{gather*}
Thus, we obtain
\begin{gather*}
\int_{\xi _1}^{\eta _1}v\pounds u-u\pounds v=-v( \eta _1)
u^{[ 1] }( \eta _1) \geq 0, \\
\int_{\xi _2}^{\eta _2}v\pounds u-u\pounds v=-v( \eta _2)
u^{[ 1] }( \eta _2) +u( \eta _2) v^{
[ 1] }( \eta _2) -u( \xi _2) v^{[ 1] }( \xi _2) \leq 0.
\end{gather*}
This completes the proof.
\end{proof}

\subsection{Spectral radius of a positive operator}

Let $Z$ be a real Banach space and $L( Z) $ the Banach space of
linear continuous operators from $Z$ into $Z$. For $L\in L( Z) $,
$r( L) =\lim \| L^{n}\| ^{\frac{1}{n}}$ denotes
the spectral radius of $L$.

A nonempty closed convex subset $K$ of $Z$ is said to be an ordered cone if
$( tK) \subset K$ for all $t\geq 0$ and $K\cap ( -K)
=\{ 0\} $. Moreover if $Z=\overline{K-K}$, the cone $K$ is said
to be total. It well known that an ordered cone $K$ induces a partial order
on the Banach space $Z\ $($x\leq y$ if and only if $y-x\in K$ for all $x,y\in Z$).

Let $K$ be an ordered cone of $Z$ and $L\in L( Z)$. $L$ is said
to be positive if $L( K) \subset K$ and $\mu \in\mathbb{R}$ is said to
be a positive eigenvalue of $L$ if there exists
$u\in K\setminus \{ 0\} $ such that $Lu=\mu u$.

Let $L_1,L_2\in L( Z) $ be two positive operators. We write
$L_1\leq L_2$ if $L_1u\leq L_2u$ for all $u\in K$.

We will use in this work the following result known as the Krein-Rutman
Theorem.

\begin{theorem}[{\cite[Proposition 7.26]{h6}}] \label{kr}
Assume that the cone $K$ is total and $L\in L( Z) $ is
compact and positive with $r( L) >0$. Then $r( L) $ is
a positive eigenvalue of $L$.
\end{theorem}

We will use also the following lemma.

\begin{lemma}[{\cite[Corollary 7.28]{h6}}] \label{lemmeXX}
Assume that the cone $K$ is total and let $L_1,L_2$ in $L( Z) $
be two compact and positive operators. If $L_1\leq L_2 $, then $r(L_1)\leq r(L_2)$.
\end{lemma}

Next we recall a fundamental result proved by Nussbaum
in \cite{nussbaum} and used in \cite{BGK}.

\begin{lemma}\label{radius}
Let $( L_n) $ be a sequence of compact linear
operators on a Banach space $Z$ and suppose that $L_n\to L$ in
operator norm as $n\to \infty $. Then $r( L_n)\to r( L) $.
\end{lemma}

\subsection{The linear eigenvalue bvp in the integrable case}

\begin{theorem}\label{Preli4}
Assume that Hypotheses \eqref{hyp1} and \eqref{hyp2} hold and
$q\in K_{G}\cap L^{1}[ \xi ,\eta ] $. Then the set of eigenvalues
to bvp \eqref{bvp3} consists of an increasing sequence of simple eigenvalues
$(\mu _k(q))_{k\geq 1}$ tending to $+\infty $, such that for all $k\geq 1$,
\begin{enumerate}
\item The eigenfunction $\phi _k$ associated with $\mu _k(q)$ belongs to
$S_k$.

\item If $\theta >0$ then $\mu _k(\theta q)=\frac{\mu _k(q)}{\theta }$.

\item Let $q_1$ be a nonnegative function in $L^{1}[ \xi ,\eta ]
$ which does not vanish identically in $[ \xi ,\eta ] $. We have$
\ \mu _k(q_1)\geq \mu _k(q)$ for all $k\geq 1$ whenever $q_1\leq q$.
Moreover, if $q_1<q$ in a subset of a positive measure then $\mu
_k(q_1)>\mu _k(q)$.

\item If $[ \xi _1,\eta _1] \subsetneq [ \xi ,\eta] $ then
$\mu _k(q,[ \xi ,\eta ] )<\mu _k(q,[ \xi_1,\eta _1] )$.

\item $\mu _k$ is a continuous function with respect to the variable $q$
lying in $L^{1}[ \xi ,\eta ] $.
\end{enumerate}
\end{theorem}

\begin{proof}
From Theorem 4.3.2 in \cite{zettl}, the bvp \eqref{bvp3} has only
real and simple eigenvalues and they are ordered to satisfy
\[
-\infty <\mu _1<\mu _2<\dots  <\lim_{k\to \infty }\mu _k=+\infty .
\]
Moreover if $\phi _k$ is an eigenfunction of $\mu _k$, and $n_k$
denotes the number of zeros of $\phi _k$ in $( \xi ,\eta ) $,
then $\phi _k\in AC[ \xi ,\eta ]$, $p\phi _k'\in AC
[ \xi ,\eta ] $ and $n_{k+1}=n_k+1$.
Now, we have
\[
0<-\phi _k( \eta ) \phi _k^{[ 1] }( \eta
) +\phi _k( \xi ) \phi _k^{[ 1] }( \xi
) +\int_{\xi }^{\eta }p( \phi _k')
^{2}=\int_{\xi }^{\eta }\phi _k\pounds \phi _k=\mu _k\int_{\xi }^{\eta
}q\phi _k^{2}
\]
leading to $\mu _k>0$ for all $k\geq 1$.

Let $L_{q}:E\to E$ be defined by
\[
L_{q}u(t)=\int_{\xi }^{\eta }G( t,s) q(s)u( s) ds=\pounds ^{-1}( qu) ( t) .
\]
It is easy to see that $L_{q}$ is a positive operator with respect to the
total cone of nonnegative functions in $E$ and $\lambda $ is an eigenvalue
of $L_{q}$ if and only if $\lambda ^{-1}$ is an eigenvalue of bvp
\eqref{bvp3}.
 Also, the presence of eigenvalues implies that $r( L_{q})>0$.
Thus, we deduce from Theorem \ref{kr} that $r( L_{q}) $ is
the largest and positive eigenvalue of $L_{q}$, and so, we have
$\mu _1=1/r( L_{q}) $ and $n_1=0$ and for all $k\geq 2$,
$ n_k=k-1$. That is $\phi _k\in S_k$ and Assertion 1, is proved.

Assertion 2 is obvious and since $\mu _k>0$ for all $k\geq 1$. Assertion
3 follows directly from \cite[Theorem 4.9.1]{zettl}.

To prove Property 4, let $[ \xi ',\eta '] \subsetneq [ \xi ,\eta ] $, and
$\phi $ and $\psi $ be such that
\begin{gather*}
\pounds \phi =\mu _k(q)q\phi ,\quad \text{a.e. } t\in ( \xi ,\eta ), \\
au(\xi )-b\lim_{t\to \xi }p( t) u'(t)=0, \\
cu(\eta )+d\lim_{t\to \eta }p( t) u'(t)=0,
\end{gather*}
and
\begin{gather*}
\pounds \psi =\mu _k(q,[ \xi _1,\eta _1] )q\psi ,\quad
\text{a.e. } t\in ( \xi ,\eta ) , \\
au(\xi _1)-b\lim_{t\to \xi _1}p( t) u'(t)=0, \\
cu(\eta _1)+d\lim_{t\to \eta _1}p( t) u'(t)=0.
\end{gather*}

Denote by $( x_i) _{1\leq i\leq k}$ and
$( y_{j})_{1\leq j\leq k}$, respectively, the two sequences of zeros
of $\phi $ and $\psi $. There exist two integers $1\leq i_0$, $j_0\leq k$
such that one of the following two situations holds:
\begin{gather*}
\xi \leq x_{i_0-1}\leq y_{j_0-1}<y_{j_0}<x_{i_0}\leq \eta , \label{i}\\
\xi \leq x_{i_0-1}<y_{j_0-1}<y_{j_0}\leq x_{i_0}\leq \eta .
\label{ii}
\end{gather*}
Without loss of generality, suppose $\phi $ and $\psi $ are positive,
respectively, in $( x_{i_0-1},x_{i_0}) $ and $(y_{j_0-1},y_{j_0}) $ and
\eqref{i} holds. Then we
have
\[
\psi ( y_{j_0}) =0,\quad \phi ( y_{j_0}) >0,\quad
\psi ^{[ 1] }( y_{j_0}) <0,\quad
\psi ^{[ 1] }(y_{j_0-1}) \geq 0,\quad
\phi ( y_{j_0-1}) \geq 0,
\]
and
\begin{align*}
&-\psi ^{[ 1] }( y_{j_0-1}) \phi (y_{j_0-1})
+\phi ^{[ 1] }( y_{j_0-1}) \psi( y_{j_0-1}) \\
&=\begin{cases}
0 &\text{if }y_{j_0-1}=\xi \\
-\psi ^{[ 1] }( y_{j_0-1}) \phi (
y_{j_0-1}) \leq 0 &\text{if }y_{j_0-1}>\xi .
\end{cases}
\end{align*}
From which we obtain
\begin{align*}
&( \mu _k(q,[ \xi ,\eta ] )-\mu _k(q,[ \xi _1,\eta
_1] )) \int_{y_{j_0-1}}^{y_{j_0}}q\psi \phi \\
&=\int_{y_{j_0-1}}^{y_{j_0}}\psi \pounds \phi -\phi \pounds \psi \\
&=\psi ^{[ 1] }( y_{j_0}) \phi (
y_{j_0}) -\psi ^{[ 1] }( y_{j_0-1}) \phi
( y_{j_0-1}) +\phi ^{[ 1] }( y_{j_0-1})
\psi ( y_{j_0-1}) <0
\end{align*}
leading to
\[
\mu _k(q,[ \xi ,\eta ] )<\mu _k(q,[ \xi _1,\eta _1] ).
\]
Finally, Property 5 is obtained from \cite[Theorem 3.5.2]{zettl}.
\end{proof}

\subsection{Berestycki's half-eigenvalue bvp}

Let $m,\alpha $ and $\beta $ be three continuous functions on $[\xi ,\eta ]$
with $m>0$ in $[\xi ,\eta ]$ and consider the bvp,
\begin{equation}
\begin{gathered}
\pounds u=\lambda mu+\alpha u^{+}-\beta u^{-}\quad \text{in }( \xi ,\eta) , \\
au(\xi )-b\lim_{t\to \xi }p(t)u'(t)=0 \\
cu(\eta )+d\lim_{t\to \eta }p(t)u'(\eta )=0,
\end{gathered}\label{bvp6}
\end{equation}
where $\lambda $ is a real parameter.

Bvp \eqref{bvp6} is called half-linear since it is linear and positively
homogeneous in the cones $u\geq 0$ and $u\leq 0$.

\begin{definition}\label{Preli7}\rm
We say that $\lambda $ is a half-eigenvalue of \eqref{bvp6} if
there exists a nontrivial solution $( \lambda ,u_{\lambda }) $ of
\eqref{bvp6}. In this situation, $\{ (\lambda ,tu_{\lambda }),\;
t>0\} $ is a half-line of nontrivial solutions of \eqref{bvp6} and
$\lambda $ is said to be simple if all solutions $(\lambda ,v)$ of \eqref{bvp6}
 with $v$ and $u$ having the same sign on a deleted neighborhood of $\xi $
are on this half-line. There may exist another half-line of solutions $
\{ (\lambda ,tv_{\lambda }),\text{ }t>0\} $, but then we say that
$\lambda $ is simple if $u_{\lambda }$ and $v_{\lambda }$ have different
signs on a deleted neighborhood of $\xi $ and all solutions
$( \lambda ,v) $ of \eqref{bvp6} lie on these two half lines.
\end{definition}

\begin{remark} \label{XXL} \rm
Note that the position of the real parameter in the differential
equation in \eqref{bvp6} is not same as in Problem \eqref{bvp2}. Moreover,
we have Problem \eqref{bvp6} coincides with the linear eigenvalue problem
when $\alpha =\beta =0$, even though Problem \eqref{bvp2} coincides with the
linear eigenvalue problem when $\alpha =\beta $.
\end{remark}

Berestycki proved in \cite{Ber1} the following theorem.

\begin{theorem}\label{Preli8}
Assume that $p\in C^{1}[ \xi ,\eta ] $ and $p>0$ in
$[ \xi ,\eta ] $. Then the set of half eigenvalues of bvp \eqref{bvp6}
consists of two increasing sequences of simple half-eigenvalues for
bvp \eqref{bvp6} $(\lambda _k^{+})_{k\geq 1}$ and
$(\lambda _k^{-})_{k\geq 1}$, such that for all $k\geq 1$ and
$\nu =+$ or $-$, the corresponding half-lines of solutions are in
$\{ \lambda _k^{\nu }\} \times S_k^{\nu }$.
\end{theorem}

\begin{proposition}\label{beres}
Let $\alpha _1,\alpha _2,\beta _1,\beta _2\in C([ \xi ,\eta ] )$. We have
\begin{itemize}
\item If $\alpha _1\leq \alpha _2$, then $\lambda _k^{\nu }(
\alpha _1) \geq \lambda _k^{\nu }( \alpha _2)$, for
all $k\geq 1$ and $\nu =+$ or $-$.

\item If $\beta _1\leq \beta _2$, then
$\lambda _k^{\nu }( \beta_1) \geq \lambda _k^{\nu }( \beta _2)$, for all $
k\geq 1$ and $\nu =+$ or $-$.
\end{itemize}
\end{proposition}

\begin{proof}
We present the proof of the first assertion. The second one can be proved in
a similar way. Let $\phi _1,\phi _2$ be such that
\begin{gather*}
\pounds \phi _1=\lambda _k^{\nu }( \alpha _1) m\phi
_1+\alpha _1\phi _1^{+}-\beta \phi _1^{-}\quad \text{ in }
( \xi ,\eta ) , \\
a\phi _1(\xi )-b\lim_{t\to \xi }p(t)\phi _1'(t)=0, \\
c\phi _1(\eta )+d\lim_{t\to \eta }p(t)\phi _1'(\eta)=0,
\end{gather*}
and
\begin{gather*}
\pounds \phi _2=\lambda _k^{\nu }( \alpha _2) m\phi
_2+\alpha _1\phi _2^{+}-\beta \phi _2^{-}\quad \text{in }( \xi ,\eta) , \\
a\phi _2(\xi )-b\lim_{t\to \xi }p(t)\phi _2'(t)=0, \\
c\phi _2(\eta )+d\lim_{t\to \eta }p(t)\phi _2'(\eta)=0.
\end{gather*}
Note that $\phi _1,\phi _2\in S_k\cap C^{2}[ \xi ,\eta ] $
and let $[ \xi _1,\eta _1] $ be the interval given by Theorem
\ref{Preli1} for the functions $\phi _1$ and $\phi _2$. Since $\phi _1$
and $\phi _2$ have the same sign in $( \xi _1,\eta _1)$,
after simple computations we obtain
\[
( \lambda _k^{\nu }(\alpha _1)-\lambda _k^{\nu }(\alpha
_2)) \int_{\xi _1}^{\eta _1}m\phi _1\phi _2=-\int_{\xi
_1}^{\eta _1}( \alpha _1-\alpha _2) \phi _1\phi
_2+\int_{\xi _1}^{\eta _1}\phi _2\pounds \phi _1-\phi _1\pounds
\phi _2\geq 0
\]
leading to
\[
\lambda _k^{\nu }(\alpha _1)\geq \lambda _k^{\nu }(\alpha _2).
\]
This completes the proof.
\end{proof}

\begin{remark} \label{rmk3.11} \rm
Naturally one can ask, is it possible to extend Berestyki's theorem to the
case where the weight $m$, as well as $\alpha$ and $\beta$ all belong to
$L_{G}^{1}[ \xi ,\eta ]$?

This is technically difficult since a half-eigenvalue of \eqref{bvp6} is
decreasing with respect to the weight $m$ only if it is positive.
\end{remark}

\subsection{Fu\v{c}ik spectrum} \label{3.5}

Consider now the bvp,
\begin{equation}
\begin{gathered}
-u''(t)=\alpha u^{+}(t)-\beta u^{-}(t),\quad t\in ( \xi,\eta ) , \\
au(\xi )-bu'(\xi )=0, \\
cu(\eta )+du'(\eta )=0,
\end{gathered}  \label{bvp7}
\end{equation}
where $\alpha ,\beta $ are positive real parameters and
$a,b,c,d\in\mathbb{R}^{+}$ with $ac+ad+bc>0$.

The statement of the next result requires introducing the functions
$\Lambda_{a,b,c,d},\Lambda _{a,b}:( 0,+\infty ) \to (0,+\infty ) $ defined,
for $\sigma >0$, by
\[
\Lambda _{a,b,c,d}( \sigma ) =\frac{1}{\sqrt{\sigma }}
\Big( \pi
-\arcsin \Big( \sqrt{\frac{b^{2}\sigma }{a^{2}+b^{2}\sigma }}\Big)
-\arcsin \Big( \sqrt{\frac{d^{2}\sigma }{c^{2}+d^{2}\sigma }}\Big)
\Big) ,
\]
and
\[
\Lambda _{a,b}( \sigma ) =\frac{1}{\sqrt{\sigma }}\Big( \pi
-\arcsin \Big( \sqrt{\frac{b^{2}\sigma }{a^{2}+b^{2}\sigma }}\Big)\Big) .
\]
Note that $\Lambda _{a,b}=\Lambda _{a,b,1,0}=\Lambda _{1,0,a,b}$.
The sets $S_k^{+}$, $S_k^{-}$ and $S_k$ are those introduced in Section 2 for
 $p=1$.
The main goal of this subsection is to describe the set
\[
F_{s}=\{ ( \alpha ,\beta ) \in\mathbb{R}\times\mathbb{R}
:\eqref{bvp7}\text{ has a solution }\}
\]
known as the Fu\v{c}ik spectrum.

\begin{theorem}\label{fucik}
Let $S$ be the set of solutions to bvp \eqref{bvp7}. Then
$S\subset \cup _{k\geq 1}S_k$. Moreover bvp \eqref{bvp7} admits a solution
\begin{enumerate}
\item in $S_1^{+}$ if and only if $\Lambda _{a,b,c,d}( \alpha )=\eta -\xi$,

\item in $S_1^{-}$ if and only if $\Lambda _{a,b,c,d}( \beta )=\eta -\xi$,

\item in $S_{2l}^{+}$ with $l\geq 1$ if and only if
\[
\Lambda _{a,b}( \alpha ) +\Lambda _{c,d}( \beta ) +\pi
( l-1) ( \frac{1}{\sqrt{\alpha }}+\frac{1}{\sqrt{\beta }}) =\eta -\xi ,
\]

\item in $S_{2l}^{-}$ with $l\geq 1$ if and only if
\[
\Lambda _{a,b}( \beta ) +\Lambda _{c,d}( \alpha ) +\pi
( l-1) ( \frac{1}{\sqrt{\alpha }}+\frac{1}{\sqrt{\beta }}) =\eta -\xi ,
\]

\item in $S_{2l+1}^{+}$ with $l\geq 1$ if and only if
\[
\Lambda _{a,b}( \alpha ) +\Lambda _{c,d}( \alpha ) +
\frac{\pi ( l-1) }{\sqrt{\alpha }}+\frac{\pi l}{\sqrt{\beta }}
=\eta -\xi ,
\]

\item in $S_{2l+1}^{-}$ with $l\geq 1$ if and only if
\[
\Lambda _{a,b}( \beta ) +\Lambda _{c,d}( \beta ) +
\frac{\pi ( l-1) }{\sqrt{\beta }}+\frac{\pi l}{\sqrt{\alpha }}
=\eta -\xi.
\]
\end{enumerate}
\end{theorem}

\begin{proof}
First, note that $u$ is a solution to \eqref{bvp7} if and only if
$v(t)=u( ( \eta -\xi ) t+\xi ) $ is a solution to the bvp
\begin{gather*}
-v''(t)=( \eta -\xi ) ^{2}\alpha v^{+}(t)-(
\eta -\xi ) ^{2}\beta v^{-}(t),\ t\in ( 0,1) , \\
av(0)-\frac{b}{( \eta -\xi ) }v'(0)=0, \\
cv(1)+\frac{d}{( \eta -\xi ) }v'(1)=0.
\end{gather*}
Then, Assertions 1 and 2 of Theorem \ref{fucik} follow from Proposition 3.1
in \cite{benmezai}.

Now, for the sake of brevity, we prove only Assertion 3 (the others can be
proved similarly). Note that $u\in S_{2l}^{\nu }$ is a solution to
\eqref{bvp7} if and only if there exists a finite sequence
$( x_i)_{i=0}^{i=2l}$ such that
\[
\xi =x_0<x_1<\dots <x_{2l-1}<x_{2l}=\eta
\]
and
\begin{gather*}
u>0\quad \text{in $( x_{2i},x_{2i+1})$ for $i=0,\ldots ,( l-1)$}, \\
u<0\quad \text{in $( x_{2i-1},x_{2i})$ for $i=1,\ldots ,l$.}
\end{gather*}
Moreover, $u$ satisfies
\begin{gather*}
-u''(t)=\alpha u(t),\quad t\in ( \xi ,x_1) , \\
au(\xi )-bu'(\xi )=u( x_1) =0,
\end{gather*}
and for $i=1,\ldots ,( l-1)$:
\begin{gather*}
-u''(t)=\alpha u(t),\quad t\in ( x_{2i},x_{2i+1}) ,
u(x_{2i})=u(x_{2i+1})=0,
\end{gather*}
and
\begin{gather*}
-u''(t)=\beta u(t),\quad t\in ( x_{2i-1},x_{2i}) , \\
u(x_{2i-1})=u(x_{2i})=0,
\end{gather*}
and
\begin{gather*}
-u''(t)=\alpha u^{+}(t)-\beta u^{-}(t),\quad t\in (x_{2l-1},\eta ) , \\
u( x_{2l-1}) =cu(\eta )+du'(\eta )=0.
\end{gather*}
Hence,  from Assertions 1 and 2, we obtain
\begin{gather*}
\frac{1}{\sqrt{\alpha }}\Big( \pi -\arcsin \Big( \sqrt{\frac{b^{2}\alpha
}{a^{2}+b^{2}\alpha }}\Big) \Big) =x_1-\xi, \\
\frac{\pi }{\sqrt{\alpha }}=( x_{2i+1}-x_{2i}) \quad \text{for }i=1,\ldots ,( l-1), \\
\frac{\pi }{\sqrt{\beta }}=( x_{2i}-x_{2i-1})\quad \text{for }i=1,\ldots ,( l-1), \\
\frac{1}{\sqrt{\beta }}\Big( \pi -\arcsin \Big( \sqrt{\frac{d^{2}\beta }{
c^{2}+d^{2}\beta }}\Big) \Big) =( \eta -x_{2l-1}) .
\end{gather*}
Summing the above equalities, we obtain
\[
\Lambda _{a,b}( \alpha ) +\Lambda _{c,d}( \beta ) +\pi
( l-1) \Big( \frac{1}{\sqrt{\alpha }}+\frac{1}{\sqrt{\beta }}\Big) =\eta -\xi .
\]
Conversely, let $\alpha ,\beta >0$ be such that
\begin{equation}
\Lambda _{a,b}( \alpha ) +\Lambda _{c,d}( \beta ) +\pi
( l-1) \Big( \frac{1}{\sqrt{\alpha }}+\frac{1}{\sqrt{\beta }}
\Big) =\eta -\xi,  \label{F1}
\end{equation}
and let $( x_i) _{i=0}^{i=2l}$ be the sequence defined by
\begin{equation}
\begin{gathered}
x_0=\xi ,\quad x_1=\xi +\frac{1}{\sqrt{\alpha }}\Big( \pi -\arcsin \Big(
\sqrt{\frac{b^{2}\alpha }{a^{2}+b^{2}\alpha }}\Big) \Big) , \\
x_{2i}=x_{2i-1}+\frac{\pi }{\sqrt{\beta }}\quad \text{for }
i=1,\ldots ,( l-1), \\
x_{2i+1}=x_{2i}+\frac{\pi }{\sqrt{\alpha }}\quad \text{for }
i=1,\ldots ,( l-1) ,\; x_{2l}=\eta .
\end{gathered}  \label{F2}
\end{equation}
Observe that from \eqref{F2} and \eqref{F1}  we have
\[
\Lambda _{a,b,1,0}( \alpha ) =x_1-\xi \quad \text{and }
\Lambda_{1,0,c,d,}( \beta ) =\eta -x_{2l-1},
\]
that is, $1$ is the smallest eigenvalue of each of the bvps
\begin{equation}
\begin{gathered}
-u''=\alpha u\ \text{in}\ ( \xi ,x_1), \\
au( \xi ) -bu'(\xi )=u( x_1) =0,
\end{gathered}  \label{A01}
\end{equation}
and
\begin{equation}
\begin{gathered}
-u''=\beta u\ \text{in}\ ( x_{2l-1},\eta ), \\
u( x_{2l-1}) =cu( \eta ) +du'(\eta )=0.
\end{gathered}  \label{A02}
\end{equation}
Thus, we consider the function
\[
\phi ( t) =\begin{cases}
\phi _1( t) & \text{for }t\in [ \xi ,x_1], \\
\phi _{2i}( t) & \text{for }t\in [x_{2i-1},x_{2i}] ,\; i=1,\ldots ,( l-1), \\
\phi _{2i+1}( t) &\text{for }t\in [
x_{2i},x_{2i+1}] ,\ i=1,\ldots ,( l-1), \\
\phi _{2l}( t) & \text{for }t\in [x_{2l-1},\eta ],
\end{cases}
\]
where $\phi _1$ is the positive eigenfunction associated with the
eigenvalue $1$ of \eqref{A01} satisfying $\phi _1'(x_1) =-1$,
\begin{gather*}
\phi _{2i}( t) =-\frac{1}{\sqrt{\beta }}\sin \big( \sqrt{\beta }
( t-x_{2i-1}) \big) \quad \text{for }i=1,\ldots ,(l-1) , \\
\phi _{2i+1}( t) =\frac{1}{\sqrt{\alpha }}\sin \big( \sqrt{
\alpha }( t-x_{2i}) \big) \quad \text{for }i=1,\ldots,( l-1),
\end{gather*}
and $\phi _{2l}$ is the negative eigenfunction associated with the
eigenvalue $1$ of \eqref{A02} satisfying $\phi _{2l}'(x_{2l-1}) =-1$.

Thus, by simple computations we find that
\begin{gather*}
\phi _{2i-1}'( x_{2i-1}) =\phi _{2i}'(
x_{2i-1}) =-1\quad \text{for }i=1,\ldots ,l, \\
\phi _{2i}'( x_{2i}) =\phi _{2i+1}'(x_{2i}) =1\quad
\text{for }i=1,\ldots ,( l-1) , \\
\phi _{2i}''( x_{2i-1}) =\phi _{2i}''( x_{2i}) =0\quad
\text{for }i=1,\ldots ,(l-1) , \\
\phi _{2i+1}''( x_{2i}) =\phi _{2i+1}''( x_{2i+1}) =0\quad
\text{for }i=1,\ldots,( l-1),\\
\phi _1''(x_1)=\phi _{2l}''(x_{2l-1})=0.
\end{gather*}
All the above equalities make $\phi$ a function in $S_{2l}^{+}\cap C^{2}
[ \xi ,\eta ] $ satisfying bvp \eqref{bvp7}. This completes the
proof.
\end{proof}

\section{Proofs of main results}

\subsection{Auxiliary results}

Let $q\in L_{G}^{1}[ \xi ,\eta ]$. For
$\varkappa \in (\xi ,\eta ) $ we define the operators
$L_{q,\varkappa ,l}:C[ \xi,\varkappa ] \to C[ \xi ,\varkappa ] $ and
$L_{q,\varkappa ,r}:C[ \varkappa ,\eta ] \to C[\varkappa ,\eta ] $ by
\begin{gather*}
L_{q,\varkappa ,l}u( t) =\int_{\xi }^{\varkappa }G_{\varkappa
,l}( t,s) q( s) u( s) ds, \\
L_{q,\varkappa ,r}u( t) =\int_{\varkappa }^{\eta }G_{\varkappa
,r}( t,s) q( s) u( s) ds,
\end{gather*}
where
\[
G_{\varkappa ,l}( t,s)
=\Big( b+a\int_{\xi }^{\varkappa }\frac{
d\tau }{p( \tau ) }\Big) ^{-1}
\begin{cases}
\big( b+a\int_{\xi }^{s}\frac{d\tau }{p( \tau ) }\big)
\int_{t}^{\varkappa }\frac{d\tau }{p( \tau ) },
& \xi \leq s\leq t\leq \varkappa , \\[4pt]
\big( b+a\int_{\xi }^{t}\frac{d\tau }{p( \tau ) }\big)
\int_{s}^{\varkappa }\frac{d\tau }{p( \tau ) },
& \xi \leq t\leq s\leq \varkappa ,
\end{cases}
\]
and
\[
G_{\varkappa ,r}( t,s)
=\Big( d+c\int_{\xi }^{\varkappa }\frac{
d\tau }{p( \tau ) }\Big) ^{-1}
\begin{cases}
\int_{\varkappa }^{s}\frac{d\tau }{p( \tau ) }\big(
d+c\int_{t}^{\eta }\frac{d\tau }{p( \tau ) }\big) ,
&\varkappa \leq s\leq t\leq \eta , \\
\int_{\varkappa }^{t}\frac{d\tau }{p( \tau ) }\big(
d+c\int_{s}^{\eta }\frac{d\tau }{p( \tau ) }\big) , &
\varkappa \leq t\leq s\leq \eta .
\end{cases}
\]

\begin{lemma}\label{lemmex}
Assume that \eqref{hyp1} and \eqref{hyp2} hold. Then, for
every function $q\in $ $K_{G}$,
$\lim_{\varkappa \to \xi }r(L_{q,\varkappa ,l}) =0$ and
$\lim_{\varkappa \to \eta }r(L_{q,\varkappa ,r}) =0$.
\end{lemma}

\begin{proof}
We will prove that $\lim_{\varkappa \to \xi }r( L_{q,\varkappa
,l}) =0$. The other limit can be obtained similarly. We distinguish
two cases:

$\bullet$ $b\neq 0$: In this case
$q\in L^{1}[ \xi ,\frac{\xi +\eta }{2}] $ and we have
\begin{align*}
r( L_{q,\varkappa ,l})
&\leq \int_{\xi }^{\varkappa }G_{\varkappa ,l}( s,s) q( s) ds \\
&\leq \Big( b+a\int_{\xi }^{\varkappa }\frac{d\tau }{p( \tau ) }
\Big) ^{-1}\int_{\xi }^{\varkappa }\Big( b+a\int_{\xi }^{\varkappa }\frac{
d\tau }{p( \tau ) }\Big) \Big( \int_{s}^{\varkappa }\frac{
d\sigma }{p( \sigma ) }\Big) q( s) ds \\
&\leq \int_{\xi }^{\varkappa }\frac{d\sigma }{p( \sigma ) }
\int_{\xi }^{\varkappa }q( s) ds,
\end{align*}
from which we obtain that $\lim_{\varkappa \to \xi }r(
L_{q,\varkappa ,l}) =0$.

$\bullet$  $b=0$: In this case $a\neq 0$ and
$q(s)\int_{\xi }^{s}1/p( \varepsilon )\, d\varepsilon
\in L^{1}[ \xi ,(\xi+\eta)/2] $ and we have
\begin{align*}
r( L_{q,\varkappa ,l}) 
&\leq \int_{\xi }^{\varkappa }G_{\varkappa ,l}( s,s) q( s) ds \\
&\leq \int_{\xi }^{\varkappa }\Big( \int_{\xi }^{s}\frac{d\sigma }{p(
\sigma ) }\Big) q( s) ds
-\Big( \int_{\xi }^{\varkappa }
\frac{d\sigma }{p( \sigma ) }\Big) ^{-1}\int_{\xi }^{\varkappa
}\Big( \int_{\xi }^{s}\frac{d\sigma }{p( \sigma ) }\Big)^{2}q( s) ds \\
&\leq 2\int_{\xi }^{\varkappa }( q( s) \int_{\xi }^{s}\frac{
d\sigma }{p( \sigma ) }) ds
\end{align*}
leading to $\lim_{\varkappa \to \xi }r( L_{q,\varkappa,l}) =0$.
This completes the proof.
\end{proof}

\begin{lemma} \label{benmezai}
Assume that \eqref{hyp1} and \eqref{hyp2} hold and let
$\alpha ,\beta \ $be two functions in $K_{G}$. If $u$ is a nontrivial
solution of
\begin{gather*}
-( pu') '(t)=\lambda ( \alpha (t) u^{+}(t)-\beta ( t) u^{-}(t)) , \\
u( t_0) =0
\end{gather*}
with $t_0\in \{ \xi ,\eta \}$, then $t_0$ is an isolated
zero of $u$ (i.e. there exists a neighborhood $V_0$ of $t_0$ such that
$u(t)\neq 0$ for all $t\in V_0$). Moreover, we have that
$\lim_{t\to t_0}p( t) u'(t)$ exists.
\end{lemma}

\begin{proof}
We present the proof for $t_0=\xi $ the other case is similar.
Let $t_{\ast }>\xi $ be such that $u$ does not vanish identically in
$( \xi,t_{\ast }) $ and suppose that $\alpha +\beta >0$ a.e. in
$( \xi ,t_{\ast }) $ (the case $\alpha +\beta =0$  a.e. in
$( \xi,t_{\ast }) $ is obvious). For the purpose of contradiction, suppose
that there is a sequence
$( \tau _n) \subset ( \xi,t_{\ast }) $ such that
$u( \tau _n) =0$ for all $n\in \mathbb{N}$ and
$\lim \tau _n=\xi $. In this case, $u$ satisfies for all
$n\in\mathbb{N}$,
\begin{equation}
\begin{gathered}
-( pu') '(t)=\lambda ( \alpha (
t) u^{+}(t)-\beta ( t) u^{-}(t)) ,\quad \text{a. e. }
 t\in( \xi ,\tau _{n+1}) , \\
u( \xi ) =u(\tau _{n+1})=0.
\end{gathered} \label{M1}
\end{equation}
Without loss of generality, assume that $u$ is positive in
$( \tau_n,\tau _{n+1}) $ and let $\mu _{n,1}( \alpha ) $ be the
first eigenvalue given by Theorem \ref{Preli4} associated with a positive
eigenvector $\psi _{n,1}$ of
\begin{gather*}
-( p\psi ') '(t)=\mu \alpha ( t)\psi (t),\quad  t\in ( \tau _n,\tau _{n+1}) , \\
\psi ( \tau _n) =\psi (\tau _{n+1})=0.
\end{gather*}
Multiplying the differential equation in \eqref{M1} by $\psi _{n,1}$, we
obtain after two integrations
\[
0\leq u^{[ 1] }( \tau _n) \psi _{n,1}( \tau_n)
=\int_{\tau _n}^{\tau _{n+1}}( \lambda -\mu _{n,1}(
\alpha ) ) \alpha \psi _{n,1}u
\]
leading to
\begin{equation}
\lambda \geq \mu _{n,1}( \alpha ) .  \label{M2}
\end{equation}

Now, let $\mu _{n,1}^{\ast }=1/r( L_{\alpha ,\tau _{n+1},l}) $
and let $\psi _{n,1}^{\ast }$ be the associated positive eigenvector.
 $\mu _{n,1}^{\ast }$ and $\psi _{n,1}^{\ast }$ satisfy
\begin{gather*}
-( p\psi _{n,1}^{\ast \prime }) '(t)
=\mu _{n,1}^{\ast }\alpha ( t) \psi _{n,1}^{\ast }(t),\quad t\in ( \xi ,\tau
_{n+1}) \\
\psi _{n,1}^{\ast }( \xi ) =\psi _{n,1}^{\ast }(\tau _{n+1})=0.
\end{gather*}
Again, multiplying the differential equation in \eqref{M1} by
 $\psi_{n,1}^{\ast }$, we obtain after two integrations
\[
0\geq -\psi _{n,1}^{[ 1] }( \tau _n) \psi
_{n,1}^{\ast }( \tau _n) =\int_{\tau _n}^{\tau _{n+1}}(
\mu _{n,1}^{\ast }-\mu _{n,1}( \alpha ) ) \alpha \psi
_{n,1}\psi _{n,1}^{\ast }
\]
leading to
\begin{equation}
\mu _{n,1}( \alpha ) \geq \mu _{n,1}^{\ast }.  \label{M3}
\end{equation}
Thus,  from \eqref{M2}, \eqref{M3} and Lemma \ref{lemmex} we obtain the
contradiction
\[
\lambda \geq \lim \mu _{n,1}( \alpha ) \geq \lim \mu _{n,1}^{\ast
}=1/\lim r( L_{\alpha ,\tau _{n+1},l}) =+\infty .
\]
Now, suppose that $u>0$ on $( \xi ,t_{\#}) $ for some
$t_{\#}>\xi$. We have by simple integration over
$( t,t_{\#}) \subset (\xi ,t_{\#}) $
\[
p(t)u^{'}( t) -p(t_{\#})u^{'}(
t_{\#}) =\lambda \int_{t}^{t_{\#}}\alpha ( s) u(
s) ds
\]
leading to
\[
\lim_{t\to \xi }p(t)u^{'}( t)
=p(t_{\#})u'( t_{\#}) +\lim_{t\to \xi
}\int_{t}^{t_{\#}}\alpha ( s) u( s) ds.
\]
This completes the proof.
\end{proof}

Let $\alpha ,\beta \ $be two functions in
$L_{\rm loc}^{1}( \xi ,\eta ) $ such that
$\alpha ( t) \geq 0$, $\beta ( t)\geq 0$ for $t\in [ \xi ,\eta ] $ a.e.
and each of $\alpha $ and $\beta $ is positive in a subset of a positive
measure; and consider the initial-value problem
\begin{equation}
\begin{gathered}
-( pu') '(t)=\lambda ( \alpha (t) u^{+}(t)-\beta ( t) u^{-}(t)), \\
u( t_0) =\lim_{t\to t_0}p(t)u'(t)=0.
\end{gathered}  \label{ivp1}
\end{equation}

By a solution to \eqref{ivp1} we mean a function
$u\in C(\bar{I})\cap C^{1}( I) $ with
$( pu') '\in L_{\rm loc}^{1}( I) $ where $I\subset ( \xi ,\eta ) $ is
an open interval such that $t_0\in \bar{I}$ and $u$ satisfies all
equations in \eqref{ivp1}.

\begin{lemma}\label{lemma1}
Assume that \eqref{hyp1} holds and let $\alpha ,\beta$ be
two functions in $K_{G}$. Then, for all
$t_0\in [ \xi ,\eta ]$, $u\equiv 0$ is the unique solution of the
initial value problem \eqref{ivp1}.
\end{lemma}

\begin{proof}
The case $\lambda =0$ is obvious. Let $\lambda \neq 0$ and $u$ be a solution
of $( \ref{ivp1}) $ defined on some interval $[
t_0,t_{\ast }] $ with $t_{\ast }\in ( t_0,\eta ) $ (the
case $u$ defined on $[ t_{\ast },t_0] $ with
$t_{\ast }\in ( \xi ,t_0) $ can be checked similarly). Since
$L_{G}^{1}[\xi ,\eta ] \subset L_{\rm loc}^{1}( \xi ,\eta ) $ and $u$ is
continuous on $[ t_0,t_{\ast }]$, $( pu')'\in L_{\rm loc}^{1}( t_0,t_{\ast }) $.
We distinguish two cases:

$\bullet$  $t_0\in ( \xi ,\eta ) $. Let $( z_i)_{i=0}^{i=n}$ be such that
\begin{gather*}
t_0=z_0<z_1<\dots <z_n=t_{\ast },\\
k_i=| \lambda | \int_{z_i}^{z_{i+1}}\frac{d\tau }{
p( \tau ) }\int_{z_i}^{z_{i+1}}( \alpha ( \tau
) +\beta ( \tau ) ) d\tau <1.
\end{gather*}
Set for $i\in \{ 0,1,\ldots ,( n-1) \} $,
$J_i=[z_i,z_{i+1}] ,\ X_i=C( J_i) $ equipped with the
sup-norm $\| \cdot \| _{i,\infty }$ and
$T_i:X_i\to X_i$ with
\[
T_iv( t) =-\int_{z_i}^{t}\Big( \frac{\lambda }{p(
s) }\int_{z_i}^{s}( \alpha ( \tau ) v^{+}( \tau
) -\beta ( \tau ) v^{-}( \tau ) ) d\tau
\Big) ds.
\]
Let $v,w\in X_i$, we have
\begin{align*}
| T_iv( t) -T_iw( t) |
&\leq \int_{z_i}^{t}\Big( \frac{| \lambda | }{p(
s) }\int_{z_i}^{s}\alpha ( \tau ) | v^{+}(
\tau ) -w^{+}( \tau ) | d\tau \Big) ds \\
&\quad +\int_{z_i}^{t}\Big( \frac{| \lambda | }{p(
s) }\int_{z_i}^{s}\beta ( \tau ) | v^{-}(
\tau ) -w^{-}( \tau ) | d\tau \Big) ds
\end{align*}
then from \eqref{ttt}
\begin{align*}
\| T_iv-T_iw\| _{i,\infty }
&\leq | \lambda | \int_{z_i}^{z_{i+1}}\frac{ds}{p( s) }
\int_{z_i}^{s}( \alpha ( \tau ) +\beta ( \tau )
) d\tau \| v-w\| _{i,\infty } \\
&\leq k_i\| v-w\| _{i,\infty }.
\end{align*}
So, $T_i$ is a $k_i-$contraction.

For $i\in \{ 0,1,\ldots ,( n-1) \} $,  let $u_i$ be
the restriction of $u$ to the interval $J_i$. We have that $u_0$ is a
fixed point of $T_0$. Indeed, since $L_{G}^{1}[ \xi ,\eta ]
\subset L_{\rm loc}^{1}( \xi ,\eta ) $ and $u$ is continuous on
$[ t_0,t_{\ast }]$, $( pu') '\in L^{1}[ t_0,t_{\ast }] $. So integrating
the differential equation in \eqref{ivp1} over
$[ z_0,s] \subset [z_0,z_1] $ we obtain from the initial value condition
\[
p(s)u_0'(s)=-\int_{z_0}^{s}( \alpha ( \tau )
u_0^{+}( \tau ) -\beta ( \tau ) u_0^{-}( \tau
) ) d\tau
\]
from which  for all $t\in [ z_0,z_1] $ we have
\[
u_0( t) =-\int_{z_0}^{t}\Big( \frac{\lambda }{p(
s) }\int_{z_i}^{s}( \alpha ( \tau ) u_0^{+}(
\tau ) -\beta ( \tau ) u_0^{-}( \tau ) )
d\tau \Big) ds=T_0u_0( t) .
\]

Thus, the fact that $T_0$ is a contraction and the trivial function is a
fixed point of $T_0$ lead to $u_0\equiv 0$, and in particular, we have
\begin{equation}
u_0( z_1) =u_0^{[ 1] }( z_1)=u_1( z_1) =u_1^{[ 1] }( z_1) =0.
\label{*1}
\end{equation}
From this equality, $u_1$ is a fixed of $T_1$.
Then for the same reasons $u_1\equiv 0$, and in particular,
\[
u_1( z_2) =u_1^{[ 1] }( z_2)
=u_2( z_2) =u_2^{[ 1] }( z_2) =0.
\]
Repeating the above process, we obtain that $u_i\equiv 0$ for all
$i\in \{ 0,1,\ldots ,( n-1) \}$; that is, $u\equiv 0$ on $[ t_0,t_{\ast }] $.

$\bullet$ $t_0=\xi $: In this case by Lemma \ref{benmezai} we can
suppose that $u>0$ in $( \xi ,t^{\ast }) $. We distinguish two
cases:

\textbf{(a)} $\alpha \in L^{1}[ \xi ,t^{\ast }] $. Let $t_{+}\in
( \xi ,t^{\ast }) $ be such that
\[
k_{+}=| \lambda | \int_{\xi }^{t_{+}}\frac{d\tau }{
p( \tau ) }\int_{\xi }^{t_{+}}\alpha ( \tau ) d\tau <1.
\]
Set $J_{+}=[ \xi ,t_{+}] ,\ X_{+}=C( J_{+}) $ equipped
with the sup-norm $\| \cdot \| _{+,\infty }$ and
$T_{+}:X_{+}\to X_{+}$ with
\[
T_{+}v( t) =-\int_{\xi }^{t}\Big( \frac{\lambda }{p(
s) }\int_{\xi }^{s}\alpha ( \tau ) v( \tau )
d\tau \Big) ds.
\]
It is easy to see that $T_{+}$ is a $k_{+}$-contraction and $u_{+}$ the
restriction of $u$ to $[ \xi ,t_{+}] $ is a fixed point of
$T_{+}, $ so $u_{+}\equiv 0$ in $[ \xi ,t_{+}] $, and in
particular, $u( t_{+}) =u^{[ 1] }( t_{+})=0$.
Thus, we conclude from the above step that $u\equiv 0$ on its interval
of definition contradicting the beginning of this step.

\textbf{(b)} $\alpha \notin L^{1}[ \xi ,t^{\ast }] $:
 In this case $b=0$ and $\int_{\xi }^{t^{\ast }}( \alpha ( t) \int_{\xi
}^{t}\frac{ds}{p(s)}) dt<\infty $. Thus, let $t_{\infty }\in (
\xi ,t^{\ast }) $ be such that
\[
k_{\infty }=| \lambda | \int_{\xi }^{t_{\infty }}\Big(
\alpha ( t) \int_{\xi }^{t}\frac{ds}{p(s)}\Big) dt<1.
\]
Set
\[
L_{\alpha }^{1}[ \xi ,t_{\infty }]
=\big\{ v:( \xi ,t_{\infty }) \to\mathbb{R} \text{ measuable and }
\int_{\xi }^{t_{\infty }}\alpha (s) | v(s)| ds<\infty \}
\]
equipped with the norm
\[
\| v\| _{L_{\alpha }^{1}[ \xi ,t_{\infty }]
}=\int_{\xi }^{t_{\infty }}\alpha ( s) | v(s)|\, ds.
\]
Let $u_{\infty }$ be the restriction of $u$ to the interval
$[ \xi,t_{\infty }]$. We claim that $u_{\infty }$ belongs to
$L_{\alpha }^{1}[ \xi ,t_{\infty }] $. Indeed, integrating the differential
equation in \eqref{ivp1} over
$[ \epsilon ,t_{\infty }] \subset ( \xi ,t_{\infty }] $, we obtain
\begin{equation}
u_{\infty }^{[ 1] }( \epsilon ) -u_{\infty }^{[ 1
] }( t_{\infty }) =\lambda \int_{\epsilon }^{t_{\infty
}}\alpha ( s) u_{\infty }( s) ds.  \label{*2}
\end{equation}
Letting $\epsilon \to \xi $ in \eqref{*2}, we obtain
\[
| \lambda | \int_{\xi }^{t_{\infty }}\alpha (
s) u_{\infty }( s) ds=| u_{\infty }^{[ 1] }( t_{\infty }) | ,
\]
and then
\[
\int_{\xi }^{t_{\infty }}\alpha ( s) | u_{\infty }(
s) | ds=\int_{\xi }^{t_{\infty }}\alpha ( s)
u_{\infty }( s) ds=| \frac{u_{\infty }^{[ 1]
}( t_{\infty }) }{\lambda }| <\infty .
\]
Now, let $T_{\infty }:L_{\alpha }^{1}[ \xi ,t_{\infty }]
\to L_{\alpha }^{1}[ \xi ,t_{\infty }] $ be defined by
\[
T_{\infty }v( t) =\int_{\xi }^{t}\Big( -\frac{\lambda }{p(
s) }\int_{\xi }^{s}\alpha ( \tau ) v( \tau )
d\tau \Big) ds.
\]
It is easy to see that $T_{\infty }$ is a $k_{\infty }-$contraction and
$u_{\infty }$ is a fixed point of $T_{\infty }$, so $u_{\infty }\equiv 0$ in
$[ \xi ,t_{\infty }] $, and in particular,
$u( t_{\infty}) =u^{[ 1] }( t_{\infty }) =0$. Thus, we
conclude from the above step that $u\equiv 0$ on its interval of definition
contradicting the beginning of this step.
This completes the proof.
\end{proof}

\begin{lemma}\label{zero}
Assume that \eqref{hyp1} and \eqref{hyp2} hold. If $\lambda $ is
a half-eigenvalue of bvp \eqref{bvp2} associated with an eigenvector $u$
then $u\in S_k$ for some $k\geq 1$.
\end{lemma}

\begin{proof}
If $u( t_0) =u^{[ 1] }( t_0) =0$ for
some $t_0\in [ \xi ,\eta ]$ then we have from Lemma \ref{lemma1}
that $u\equiv 0$, contradicting $( \lambda ,u) $ is a
nontrivial solution of \eqref{bvp2}. This shows that $u$ has only simple
zeros.

Now, to the contrary, assume that $u$ has an infinite sequence of
consecutive zeros $( t_n) $ converging to some
$t^{\ast }\in [ \xi ,\eta ] $. We have from the continuity of
$u$, $u(t^{\ast }) =0$ and so from Lemma \ref{benmezai}
$t^{\ast }\in (\xi ,\eta ) $. Because of the simplicity of zeros
of $u$, we have that
$( t_n) =( t_n^{1}) \cup ( t_n^{2}) $ with $u^{[ 1] }( t_n^{1}) >0$ and
$u^{[ 1] }( t_n^{2}) <0$. Since
$u\in C^{1}[t^{\ast }-\varepsilon ,t^{\ast }-\varepsilon ] $ for some
$\varepsilon>0$ small enough, we obtain that
\[
0\leq \lim u^{[ 1] }( t_n^{1}) =u^{[ 1]
}( t^{\ast }) =\lim u^{[ 1] }( t_n^{2})
\leq 0.
\]
Again by Lemma \ref{lemma1}, $u\equiv 0$, contradicting
$( \lambda,u) $ is a nontrivial solution of \eqref{bvp2}.
\end{proof}

\begin{lemma}\label{lemmex1}
Assume that \eqref{hyp1} and \eqref{hyp2} hold and
$\alpha,\beta \in K_{G}\cap L^{1}[ \xi ,\eta ] $. Then, for each integer
$k\geq 1$ and $\nu =+$ or $-$, bvp \eqref{bvp2} admits at most one simple
half-eigenvalue having an eigenvector in $S_k^{\nu }$.
\end{lemma}

\begin{proof}
To the contrary, suppose that
$( \lambda _i,\phi _i) \in\mathbb{R}\times S_k^{\nu }$
satisfy \eqref{bvp2} for $i=1,2$. Then the
integrability of $1/p,\alpha $ and $\beta $ implies that
$\phi _i\in S_k^{\nu }\cap AC[ \xi ,\eta ] $ and
$p\phi _i'\in AC [ \xi ,\eta ] $. Let $[ \xi _1,\eta _1] $ and
$[ \xi _2,\eta _2] $ be the intervals given by
Theorem \ref{Preli2}. Since $\phi _1$ and $\phi _2$ have the same
sign in each of $[ \xi _1,\eta _1] $ and $[ \xi _2,\eta _2]$,
we have
\begin{gather*}
0\leq \int_{\xi _1}^{\eta _1}\phi _2\pounds \phi _1-\phi _1\pounds
\phi _2=( \lambda _1-\lambda _2) \int_{\xi _1}^{\eta
_1}\alpha \phi _1^{+}\phi _2^{+}+\beta \phi _1^{-}\phi _2^{-},
\\
0\geq \int_{\xi _2}^{\eta _2}\phi _2\pounds \phi _1-\phi _1\pounds
\phi _2=( \lambda _1-\lambda _2) \int_{\xi _2}^{\eta
_2}\alpha \phi _1^{+}\phi _2^{+}+\beta \phi _1^{-}\phi _2^{-},
\end{gather*}
leading to $\lambda _1=\lambda _2$.

Now, suppose that $\lambda $ is a half-eigenvalue of \eqref{bvp2} having two
eigenvectors $\phi _1$ and $\phi _2$ with $\phi _1\phi _2>0$ in a
right neighborhood of $\xi $,
 $\phi _1,\phi _2\in AC[ \xi ,\eta ] $ and
 $p\phi _1',p\phi _2'\in AC[ \xi,\eta ] $. Because of the positive
homogeneity of bvp \eqref{bvp2},
there exists two eigenvectors $\psi _1$ and $\psi _2$ associated with
$\lambda $ such that $\psi _1\psi _2>0$ in a right neighborhood of
$\xi $, $\psi _1,\psi _2\in AC[ \xi ,\eta ] $,
$p\psi _1',p\psi _2'\in AC[ \xi ,\eta ] $ and
\[
\psi _1( \xi ) =\psi _2( \xi ) =b,\quad
\psi _1^{[ 1] }( \xi ) =\psi _2^{[ 1]}( \xi ) =a.
\]
Indeed; Without loss of generality, suppose that $\phi _1>0$ and
$\phi_2>0$ in a right neighborhood of $\xi $. Then we distinguish the following
three cases.

$\bullet$  $\phi _1(\xi )=0$.
 In this case we have $b=0$ and from \eqref{hyp2}
 that $a>0$ (otherwise if $b\neq 0$ we obtain from the boundary condition at
$\xi $ that $\phi _1^{[ 1] }( \xi ) =0$ and Lemma \ref{lemma1} leads
to $\phi _1=0$). The positivity of $\phi _1$ near $\xi $
leads to $\phi _1^{[ 1] }( \xi ) >0$. Since $a>0$,
$b=0$ and $\phi _2>0$ near $\xi $, we have $\phi _2(\xi )=0$ and
$\phi_2^{[ 1] }( \xi ) >0$.
Thus,
\[
\psi _1=\frac{a\phi _1}{\phi _1^{[ 1] }( \xi
) },\quad \psi _2=\frac{a\phi _2}{\phi _2^{[ 1] }(
\xi ) }
\]
are eigenvectors associated with $\lambda $ satisfying
\[
\psi _1( \xi ) =\psi _2( \xi ) =b\quad \text{and}\quad
\psi _1^{[ 1] }( \xi ) =\psi _2^{[ 1]}( \xi ) =a.
\]

$\bullet$  $\phi _1^{[ 1] }( \xi ) =0$. In this case we
have $a=0$ and from \eqref{hyp2} that $b>0$ (otherwise if $a\neq 0$ we obtain
from the boundary condition at $\xi $ that $\phi _1(\xi )=0$ and Lemma
\ref{lemma1} leads to $\phi _1=0$). The positivity of $\phi _1$ near $\xi $
leads to $\phi _1( \xi ) >0$. Since $b>0$, $a=0$ and
$\phi_2>0$ near $\xi $, we have $\phi _2^{[ 1] }( \xi )=0$ and
$\phi _2(\xi )>0$.
Thus,
\[
\psi _1=\frac{b\phi _1}{\phi _1( \xi ) },\quad
\psi_2=\frac{b\phi _2}{\phi _2( \xi ) }
\]
are eigenvectors associated with $\lambda $ satisfying
\[
\psi _1( \xi ) =\psi _2( \xi ) =b\quad \text{and}\quad
\psi _1^{[ 1] }( \xi ) =\psi _2^{[ 1]}( \xi ) =a.
\]

$\bullet$ $\phi _1( \xi ) >0$ and $\phi _1^{[ 1]}( \xi ) >0$.
This happens only in the case $a>0$ and $b>0$ and
we have the boundary condition at $\xi , \phi _1( \xi ) >0$
and $\phi _1^{[ 1] }( \xi ) >0$. Thus,
\[
\psi _1=\frac{a\phi _1}{\phi _1^{[ 1] }( \xi ) }
=\frac{ b\phi _1}{\phi _1( \xi ) },\quad
\psi _2=\frac{a\phi _2}{\phi _2^{[ 1] }( \xi ) }
=\frac{b\phi _2}{\phi_2( \xi ) }
\]
are eigenvectors associated with $\lambda $ satisfying
\[
\psi _1( \xi ) =\psi _2( \xi ) =b\ \text{and}\
\psi _1^{[ 1] }( \xi ) =\psi _2^{[ 1]
}( \xi ) =a.
\]


At this stage, $\psi =\psi _1-\psi _2$ satisfies
\begin{gather*}
-( p\psi ') '(t)=\lambda ( \alpha (
t) \psi ^{+}(t)-\beta ( t) \psi ^{-}(t)) \\
\psi ( \xi ) =\psi ^{[ 1] }(\xi )=0,
\end{gather*}
and we have from Lemma \ref{lemma1}, $\psi =0$. That is,
$\psi _1=\psi _2 $, and then $\phi _1=\omega \phi _2$ with $\omega >0$.
This shows that the half-eigenvalue $\lambda $ is simple and completes
 the proof of Lemma \ref{lemmex1}.
\end{proof}

For $q\in K_{G}$ we define the linear compact operator
$L_{q}:E\to E $ by
\[
L_{q}u(t)=\int_{\xi }^{\eta }G( t,s) q( s) u(s)ds.
\]
Since, we will use the global bifurcation theorem of Rabinowitz to prove the
main result of this paper, we need to discuss the geometric and algebraic
multiplicities of characteristic values of $L_{q}\ $(which are also
eingenvalues of bvp \eqref{bvp3}).
Let $\mu _0$ be a characteristic
value of $L_{q}$ and note that
$N( \mu _0L_{q}-I) \subset N( \mu _0L_{q}-I) ^{2}$.
Thus, if $\mu _0$ is not simple then $\mu _0$ is of algebraic multiplicity
greater than $1$. We know from
Theorem \ref{Preli4} that if $q\in K_{G}\cap L^{1}[ \xi ,\eta ] $
then all characteristic values of $L_{q}$ have the geometric multiplicity
equal to one, so let us see what can happens with the algebraic mutiplicity.

\begin{lemma}\label{multalgintegrable}
Assume that \eqref{hyp1} and \eqref{hyp2} hold and
$q\in K_{G}\cap L^{1}[ \xi ,\eta ] $. Then, all characteristic
values of $L_{q}$ are of algebraic multiplicity one.
\end{lemma}

\begin{proof}
Let $( \mu _k( q) ) $ be the sequence of
characteristic values of $L_{q}$ given by Theorem \ref{Preli4}. Thus the
eigenvector $\phi _k$ associated with $\mu _k( q) $ satisfies
\begin{gather*}
-( p\phi _k') '(t)=\mu _k( q)
q( t) \phi _k(t),\quad \text{a.e. }t\in ( \xi,\eta ) , \\
a\phi _k(\xi )-b\lim_{t\to\xi} p(t)\phi_k'(t)=0, \\
c\phi _k(\eta )+d\lim_{t\to\eta} p(t)\phi_k'(t)=0.
\end{gather*}
 Multiplying by $\phi _k$ and integrating over $[ \xi ,\eta ] $
we obtain
\begin{equation}
\int_{\xi }^{\eta }p( \phi _k') ^{2}=\mu _k(q) \int_{\xi }^{\eta }
q\phi _k^{2}  \label{s+32}
\end{equation}
leading to
\begin{equation}
\mu _k( q) >0, \quad\text{and}\quad \int_{\xi }^{\eta }q\phi ^{2}>0.
\label{S10}
\end{equation}
Now, let $u\in N\big( ( \mu _k( q) L_{q}-I)
^{2}\big) $ and set $v=( \mu _k( q) L_{q_0}-I)( u) =\mu _k( q) L_{q}u-u$.
We have $\mu _k(q) L_{q}v-v=0$ leading to $v=x\phi $ and
\begin{equation}
\mu _k( q) L_{q}u-u=x\phi _k.  \label{S0}
\end{equation}
On the other hand we have that $u$ satisfies the bvp
\begin{equation}
\begin{gathered}
-( pu') '(t)=\mu _k( q) q(
t) u(t)-x\mu _k( q) q( t) \phi _k(
t) ,\quad \text{a.e.  }t\in ( \xi ,\eta ) , \\
au(\xi )-b\lim_{t\to\xi} p(t)u'(t)=0, \\
cu(\eta )+d\lim_{t\to\eta} p(t)u'(t)=0.
\end{gathered}  \label{S2}
\end{equation}
Multiplying the differential equation in \eqref{S2} by $\phi _k$ and
integrating on $( \xi ,\eta ) $ we obtain
\[
x\mu _k( q) \int_{\xi }^{\eta }q\phi _k^{2}=0.
\]

Because of \eqref{s+32}, the above equality leads to $x=0$.
Therefore, we obtain from \eqref{S0} that
$u=\omega \phi _k\in N( \mu _k(q) L_{q}-I) $ with
$\omega \in\mathbb{R}$. This completes the proof.
\end{proof}

It remains to discuss the geometric and algebraic multiplicities of
characteristic values of $L_{q}$ when
$q\in ( K_{G}\setminus L^{1}[\xi ,\eta ] ) $. We need the following
 lemma which is a version of L'Hopital's rule.

\begin{lemma}\label{addedlemma}
Let $f$ and $g$ be two differentiables functions on
$( \xi ,\xi +\epsilon ) $ with $\epsilon >0$ such that
$\lim_{t\to \xi }f( t) =\lim_{t\to \xi }g(t) =+\infty $. If
$\lim_{t\to \xi }\frac{f'(t) }{g'( t) }=l$ then
$\lim_{t\to \xi } \frac{f( t) }{g( t) }=l$.
\end{lemma}

\begin{lemma}\label{lemmeAB}
Assume that \eqref{hyp1} and \eqref{hyp2} hold and
$q\in K_{G}\setminus L^{1}[ \xi ,\eta ] $. Let $\mu $ be a
characteristic value of $L_{q}$ associated with an eigenvector $\phi $. We
have
\begin{enumerate}
\item If $\phi $ does not change sign then $\mu $ is double

\item If $\phi $ has more than one zero in $( \xi ,\eta ) $ then $
\mu $ is simple.
\end{enumerate}
\end{lemma}

\begin{proof}
Suppose that $\psi $ is another eigenvector associated with the
caracteristic value $\mu$ and let $W=W( \phi ,\psi ) $ be the
Wronksian of $\phi $ and $\psi $. By simple computations follows
$( pW) '=0$, from which we obtain
\begin{equation}
\phi \psi '-\phi '\psi =\frac{B}{p},\quad B\in\mathbb{R}.  \label{me}
\end{equation}
Considering \eqref{me} as a linear first order differential equation where
the unknown is $\psi $, we obtain that $\psi$ takes the form
\begin{gather*}
\psi ( t) = A\phi ( t) +B\psi _{\varepsilon }(t) \quad \text{with }
A,B\in\mathbb{R},\ \varepsilon \in ( \xi ,\eta ) , \\
\psi _{\varepsilon }( t) =\phi( t) \int_{\varepsilon }^{t}\frac{ds}{p( s) \phi
^{2}( s) }.
\end{gather*}

Thus, we have to examine for $\varepsilon \in ( \xi ,\eta ) $
the ability of the function $\psi _{\varepsilon }$ to be an eigenvector
associated with $\mu $ or not. Without loss of generality, suppose that $q$
is not integrable at $\xi $ and $\eta $ (the other cases can be checked
similarly). This occurs if $b=d=0$ and in this case the boundary
conditions in bvp \eqref{bvp1} become the Dirichlet conditions
\begin{equation}
u( \xi ) =u( \eta ) =0.  \label{bc}
\end{equation}

1. Suppose that $\phi $ is positive in $( \xi ,\eta ) $ and
let $\varepsilon \in ( \xi ,\eta ) $ be fixed. We have by
simple computations
\begin{equation}
p(t)\psi _{\varepsilon }'( t) =\frac{1}{\phi (
t) }+p(t)\phi '( t) \int_{\varepsilon }^{t}\frac{ds
}{p( s) \phi ^{2}( s) },\quad \text{for all }
t\in (\xi ,\eta )  \label{+++}
\end{equation}
then
\begin{equation}
-( p\psi _{\varepsilon }') '( t)
=\lambda q( t) \psi _{\varepsilon }( t) ,\quad \text{a.e. }
 t\in ( \xi ,\eta ) .  \label{---}
\end{equation}
Moreover, since $q$ is not integrable at $\xi $ and $\eta $,  from
Lemma \ref{benmezai} and \cite[Theorem 2.3.1]{zettl} we have
\[
\lim_{t\to \xi }p(t)\phi '( t) =\infty, \quad
\lim_{t\to \eta }p(t)\phi '(t) =\infty .
\]
Thus,  from Lemma \ref{addedlemma} when $\lim_{t\to \xi
}\int_{\varepsilon }^{t}\frac{ds}{p( s) \phi ^{2}( s) }
=\infty \ $(the case $\int_{\xi }^{\varepsilon }\frac{ds}{p( s)
\phi ^{2}( s) }<\infty $ is obvious), we have
\[
\lim_{t\to \xi }\psi _{\varepsilon }( t)
=\lim_{t\to \xi }\frac{\big( \int_{\varepsilon }^{t}\frac{ds}{
p( s) \phi ^{2}( s) }\big) '}
{( \frac{1}{\phi ( t) }) '}=\lim_{t\to \xi }\frac{
\frac{1}{p( t) \phi ^{2}( t) }}{-\frac{\phi '( t) }{\phi ^{2}( t) }}
=-\frac{1}{\lim_{t\to \xi }p(t)\phi '( t) }=0
\]
and also
\[
\lim_{t\to \eta }\psi _{\varepsilon }( t) =-\frac{1}{
\lim_{t\to \eta }p(t)\phi '( t) }=0.
\]
That is, $\psi _{\varepsilon }$ satisfies the boundary conditions \eqref{bc}
and all the above shows that $\psi _{\varepsilon }$ is an eigenvector of the
characteristic value $\mu $ of $L_{q}$. Moreover, since the function $
\varphi _{\varepsilon }( t) =A+B\int_{\varepsilon }^{t}\frac{ds}{
p( s) \phi ^{2}( s) }0$ vanishes at most once$\ $in $
( \xi ,\eta ) $, the eigenvector $\psi $ lies in $S_1\cup
S_2\ $and this shows that $\mu $ is double.
\smallskip

2.  Note that if $\phi ( t_1) =0$ for some $t_1\in (
\xi ,\eta ) $, we obtain from \eqref{+++} following $t_1>\varepsilon
$ and $t_1<\varepsilon $ that at least one of the limits
\[
\lim_{t>t_1,\, t\to t_1} p(t)\psi _{\varepsilon}'( t) ,\quad
\lim_{t<t_1,\, t\to t_1}p(t)\psi _{\varepsilon }'( t)
\]
are infinite and this means that $\psi _{\varepsilon }\notin Y_{\#}$. So, the
function $\psi _{\varepsilon }$ can not be an eigenvector associated with
the characteristic value $\mu $ of $L_{q}$.

By the contrary suppose that the characteristic value $\mu $ is not simple
and there exists another eigenvector of $\mu $, $\phi _1\in Y_{\#}$. In
this case, arguing as in the discussion in the beginning of this proof we obtain
\[
\phi ( t) =A\phi _1( t) +B\phi _1( t)
\int_{\varepsilon }^{t}\frac{ds}{p( s) \phi _1^{2}(s) }
\]
and
\[
p(t)\phi '( t) =A\phi _1( t) +\frac{B}{\phi
_1( t) }+p(t)\phi _1'( t)
\int_{\varepsilon }^{t}\frac{ds}{p( s) \phi _1^{2}(
s) },\quad \text{for all }t\in ( \xi ,\eta )
\]
where $A,B\in\mathbb{R}$ and $\varepsilon \in ( \xi ,\eta ) $.

Thus, arguing as in the beginning of part 2 of this proof we obtain that the
eigenvector $\phi _1$ must be positive. Therefore, from
\[
\phi ( t) =\phi _1( t) \Big( A+B\int_{\varepsilon
}^{t}\frac{ds}{p( s) \phi _1^{2}( s) }\Big)
\]
yields that $\phi $ has at most one zero in $( \xi ,\eta ) $.
Contradicting $\phi $ has more than one zero.
This completes the proof.
\end{proof}

\begin{lemma}\label{simplicitealgebrique}
Assume that \eqref{hyp1} and \eqref{hyp2} hold
and $q\in K_{G}\setminus L^{1}[ \xi ,\eta ] $. If $\mu $ is a
characteristic value of $L_{q}$ associated with an eigenvector $\phi $
vanishing more than one time in $( \xi ,\eta ) $ then $\mu$ is
of algebraic multiplicity one.
\end{lemma}

\begin{proof}
In the same way as in the proof of Lemma \ref{multalgintegrable}, 
let us show that if $u$ is a
solution to
\begin{equation}
\begin{gathered}
-( pu') '(t)=\mu q( t) u(t)-x\mu
q( t) \phi ( t) ,\quad \text{a.e. } t\in ( \xi ,\eta) , \\
au(\xi )-b\lim_{t\to\xi} p(t)u'(t)=0, \\
cu(\eta )+d\lim_{t\to\eta} p(t)u'(t)=0.
\end{gathered}  \label{S20}
\end{equation}
then $u=\omega \phi $ with $\omega \in\mathbb{R}$.
Let $u$ be a solution to \eqref{S20} and $W=W( \phi ,u) $ be
the Wronksian of $\phi $ and $u$. We have that $W$ satisfies
\[
( pW) '=( pu'\phi -pu\phi ') '=x\mu q\phi ^{2}
\]
leading to
\begin{equation}
u'\phi -u\phi '=\frac{B}{p}+\frac{x\mu }{p}
\int_{\varepsilon }^{s}q( \tau ) ( \phi ( \tau )
) ^{2}d\tau ,\quad B\in\mathbb{R}.  \label{me010}
\end{equation}
Considering \eqref{me010} as a linear first order differential equation
where the unknown is $u$, we obtain that $u$ takes the form
\begin{equation}
\begin{aligned}
u(t)&=A\phi ( t) +B\phi ( t) \int_{\varepsilon }^{t}
\frac{ds}{p( s) \phi ^{2}( s) } \\
&\quad +x\mu \phi ( t) \int_{\varepsilon }^{t}\Big( \frac{1}{p(
s) \phi ^{2}( s) }\int_{\varepsilon }^{s}q( \tau
) ( \phi ( \tau ) ) ^{2}d\tau \Big) ds,
\end{aligned} \label{nbv0}
\end{equation}
for $A,B\in\mathbb{R}$ and $\varepsilon \in ( \xi ,\eta )$.

Arguing as in 2 of the proof of Lemma \ref{lemmeAB}, we see that the
expression for $u$ given in \eqref{nbv0} is a solution of \eqref{S2} if and
only if $B=x=0$. This completes the proof.
\end{proof}


\begin{lemma}\label{regularite}
Assume that \eqref{hyp1} and \eqref{hyp2} hold and let
$( q_n) $ be a sequence in $L_{G}^{1}$ converging to
$q\in L_{G}^{1}$. Then $L_{q_n}\to L_{q}$ as $n\to \infty $ in
operator norm. Moreover, for all $[ \xi _0,\eta _0] \subset( \xi ,\eta ) $
\[
p( L_{q_n}u) '\to p( L_{q}u)'\quad
\text{in $C[ \xi _0,\eta _0]$ for all }u\in E.
\]
\end{lemma}

\begin{proof}
It is easy to check that for all
$[ \xi _0,\eta _0] \subset ( \xi ,\eta ) $, $q_n\to q$ in
$L^{1}[ \xi_0,\eta _0]$, $\Phi _{ab}q_n\to \Phi _{ab}q$ in
$L^{1}[ \xi ,\xi _0] $, and $\Psi _{cd}q_n\to \Psi _{cd}q\ $
in $L^{1}[ \eta _0,\eta ] $. Thus, we have, for all
$[ \xi _0,\eta _0] \subset ( \xi ,\eta ) $ and $u\in E$ with
$\| u\| _{\infty }=1$,
\[
\sup_{t\in [ \xi ,\eta ] }| L_{q_n}u( t)
-L_{q}u( t) | \leq \int_{\xi }^{\eta }G(s,s) | q_n( s) -q(s)| ds
=\|q_n-q\| _{G},
\]
leading to $L_{q_n}\to L_{q}$ as $n\to \infty $.

Also,  for all $[ \xi _0,\eta _0] \subset ( \xi
,\eta ) $ and $u\in E$ with $\| u\| _{\infty }=1,$, we have
\begin{align*}
&| p( t) ( L_{q_n}u) '(t) -p( t) ( L_{q}u) '( t)| \\
&\leq \int_{\xi _0}^{\eta _0}\frac{c\Phi _{ab}( s)
+a\Psi _{cd}( s) }{\Delta }| q_n( s)-q(s)| ds \\
&\quad +\int_{\xi }^{\xi _0}\frac{c}{\Delta }\Phi _{ab}( s) |
q_n( s) -q(s)| ds+\int_{\eta _0}^{\eta }\frac{a}{
\Delta }\Psi _{cd}( s) | q_n( s)
-q(s)| ds \\
&=\frac{c}{\Delta }\| \Phi _{ab}( q_n-q) \|
_{L^{1}[ \xi ,\xi _0] .}+\frac{a}{\Delta }\| \Psi
_{cd}( q_n-q) \| _{L^{1}[ \eta _0,\eta ]
.}+\| q_n-q\| _{L^{1}[ \xi _0,\eta _0] }
\end{align*}
leading to $p( L_{q_n}u) '\to p(L_{q}u) '$ in $ C[ \xi _0,\eta _0]$.
The proof is complete.
\end{proof}

\begin{lemma}\label{lalimite}
Assume that \eqref{hyp1} and \eqref{hyp2} hold and let
$( \alpha _n) $ and $( \beta _n) $ be two sequences
in $K_{G}$ converging, respectively, to $\alpha $ and
$\beta$ in $L_{G}^{1}[ \xi ,\eta ] $. Assume also that for all integers
$n\geq 1$, there exist $\lambda _n>0$ and
$\phi _n\in E\setminus \{ 0\}$ satisfying
\begin{gather*}
-( p\phi _n') '(t)=\lambda _n( \alpha
_n( t) \phi _n^{+}(t)-\beta _n( t) \phi
_n^{-}(t)), \quad t\in ( \xi ,\eta ) , \\
a\phi _n(\xi )-b\lim_{t\to \xi} p(t)\phi _n'(t)=0, \\
c\phi _n(\eta )+d\lim_{t\to \eta} p(t)\phi_n'(t)=0.
\end{gather*}
We have $\phi _n=\lambda _nA_n\phi _n$ where
$A_n=L_{\alpha_n}I^{+}-L_{\beta _n}I^{-}$.

If $( \lambda _n) $ converges to $\widetilde{\lambda }>0$,
then there exists $\psi \in E\setminus \{ 0\} $ such that $\psi =
\widetilde{\lambda }A\psi $\ where $A=L_{\alpha }I^{+}-L_{\beta }I^{-}$
(i.e. $\widetilde{\lambda }$ is a half-eigenvalue to bvp \eqref{bvp2}).
\end{lemma}


\begin{proof}
First, note that Lemma \ref{regularite} guarantee that $L_{\alpha
_n}\to L_{\alpha }$ and $L_{\beta _n}\to L_{\beta }$ in
operator norm. Let $\phi _n$ be the eigenvector corresponding to $\lambda
_n$ with $\| \phi _n\| _{\infty }=1$ and set $\psi
_n=\lambda _nA\phi _n$ and $\psi =\lim \psi _n$ (up to a
subsequence). We have
\begin{align*}
\| \phi _n-\psi \| _{\infty }
&=\| \lambda _nA_n( \phi _n) -\psi \| _{\infty }\\
&\leq |\lambda _n| \| A_n( \phi _n) -A(\phi _n) \| _{\infty }
 +\| \lambda _nA( \phi_n) -\psi \| _{\infty } \\
&\leq | \lambda _n| \| L_{\alpha_n}-L_{\alpha }\| 
 +| \lambda _n| \|L_{\beta _n}-L_{\beta }\| 
 +\| \lambda _nA( \phi_n) -\psi \| _{\infty },
\end{align*}
leading to $\lim \phi _n=\psi $ and $\| \psi \| _{\infty}=1$.
Also we have
\begin{align*}
\| \lambda _nA_n( \phi _n) -\widetilde{\lambda}A( \psi ) \| _{\infty }
&\leq \| \lambda _nA_n( \phi _n) -\lambda _nA_n( \psi )
 \| _{\infty }+\| \lambda _nA_n( \psi )
 -\lambda _nA( \psi ) \| _{\infty } \\
&\quad +\| \lambda _nA( \psi ) -\widetilde{\lambda }
 A( \psi ) \| _{\infty } \\
&\leq | \lambda _n| \| A_n\|\| \phi _n-\psi \| _{\infty }
 +| \lambda _n| \| A_n-A\| +| \lambda _n-
 \widetilde{\lambda }| \| A\| ,
\end{align*}
leading to $\lim \lambda _nA_n( \phi _n) =\widetilde{
\lambda }A( \psi )$.
At the end, letting $n\to \infty $ in the equation
$\phi _n=\lambda _nA_n( \phi _n) $ we obtain
$\psi =\widetilde{\lambda }A\psi $.
\end{proof}

\begin{remark} \label{regularite1} \rm
Arguing as in the proof of Lemma \ref{regularite}, one can prove that
$p\phi _n'\to p\psi '$ in $C[ \xi _0,\eta_0]$ for all
$[ \xi _0,\eta _0] \subset( \xi ,\eta )$
where $\phi _n$ and $\psi $ are those of the above proof.
\end{remark}

Let $\Lambda _{a,b,c,d}$ and $\Lambda _{a,b}$ be the functions defined in
Subsection \ref{3.5}. We deduce from Theorem \ref{fucik} a first result for
existence of half-eigenvalues in the case where $p\equiv 1$ and the
functions $\alpha $ and $\beta $ are constants.

\begin{corollary}\label{coro2}
Assume that $p\equiv 1\ $and $\alpha $ and $\beta $ are
positive constants. Then bvp \eqref{bvp2} admits two sequences of half
eigenvalues $( \lambda _k^{+}) $ and $( \lambda_k^{-}) $ such that
\begin{itemize}
\item $\lambda _1^{+}$ is the unique solution of $\Lambda _{a,b,c,d}(
\alpha \sigma ) =\eta -\xi $,

\item $\lambda _1^{-}$ is the unique solution of $\Lambda _{a,b,c,d}(
\beta \sigma ) =\eta -\xi $,

\item $\lambda _{2l}^{+}$ with $l\geq 1$ is the unique solution of
\[
\Lambda _{a,b}( \alpha \sigma ) +\Lambda _{c,d}( \beta
\sigma ) +\pi ( l-1) \Big( \frac{1}{\sqrt{\alpha \sigma }}+
\frac{1}{\sqrt{\beta \sigma }}\Big) =\eta -\xi ,
\]

\item $\lambda _{2l}^{-}$ with $l\geq 1$ is the unique solution of
\[
\Lambda _{a,b}( \beta \sigma ) +\Lambda _{c,d}( \alpha
\sigma ) +\pi ( l-1) \Big( \frac{1}{\sqrt{\alpha \sigma }}+
\frac{1}{\sqrt{\beta \sigma }}\Big) =\eta -\xi ,
\]

\item $\lambda _{2l+1}^{+}$ with $l\geq 1$ is the unique solution of
\[
\Lambda _{a,b}( \alpha \sigma ) +\Lambda _{c,d}( \alpha
\sigma ) +\frac{\pi ( l-1) }{\sqrt{\alpha \sigma }}+\frac{
\pi l}{\sqrt{\beta \sigma }}=\eta -\xi ,
\]

\item $\lambda _{2l+1}^{-}$ with $l\geq 1$ is the unique solution of
\[
\Lambda _{a,b}( \beta \sigma ) +\Lambda _{c,d}( \beta \sigma
) +\frac{\pi ( l-1) }{\sqrt{\beta \sigma }}+\frac{\pi l}{
\sqrt{\alpha \sigma }}=\eta -\xi .
\]
\end{itemize}
\end{corollary}



\begin{proposition}\label{thm8}
Assume that $p\equiv 1$ and $\alpha $ and $\beta $ are positive
and continuous on $[ \xi ,\eta ] $. Then the set of
half-eigenvalues of bvp \eqref{bvp2} consists of two increasing sequences of
simple half-eigenvalues $(\lambda _k^{+})_{k\geq 1}$ and
$(\lambda_k^{-})_{k\geq 1}$, such that for all $k\geq 1$ and $\nu =+$ or $-$, the
corresponding half-lines of solutions are in
$\{ \lambda _k^{\nu }\} \times S_k^{\nu }$. Moreover for all
 $k\geq 1$ and $\nu =+$ or $-,\ \lambda _k^{\nu }$ is a decreasing function
with respect to the weights $\alpha $ and $\beta $ lying in $C[ \xi ,\eta ] $.
\end{proposition}

\begin{proof}
Consider the bvp
\begin{equation}
\begin{gathered}
-u''(t)=\theta ( \alpha ( t) +\beta (
t) ) u^{+}(t)+\lambda \alpha ( t) u^{-}(t)-\lambda
\beta ( t) u^{+}(t),\ t\in ( \xi ,\eta ) , \\
au(\xi )-b\lim_{t\to\xi} p(t)u'(t)=0, \\
cu(\eta )+d\lim_{t\to\eta} p(t)u'(t)=0.
\end{gathered}  \label{HEVI}
\end{equation}
From Theorem \ref{Preli8} We have that for each integer $k\geq 1,\ \nu =+$
or $-$ and all $\lambda \geq 0$, there exists a unique
$\theta _k^{\nu }( \lambda ) $ such that \eqref{HEVI} has a
solution in $S_k^{\nu }$.

Note that $\theta _k^{\nu }( 0) =\mu _k( \alpha +\beta) >0$.
 Now we claim that there exists $\lambda _0>0$ such that
$\theta _k^{\nu }( \lambda _0) \leq \lambda _0$. To the
contrary, assume that for all $\lambda \geq 0$,
$\theta _k^{\nu }(\lambda ) >\lambda $. Thus we have from
Proposition \ref{beres} that
\[
\lambda <\theta _k^{\nu }( \lambda ) <\theta _k^{\nu }(
\lambda ,\alpha _{+},\beta _{+}) =\theta ^{\ast }( \lambda )
\]
where for $k\geq 1$ and $\nu =+$ or $-$, $\theta _k^{\nu }( \lambda
,\alpha _{+},\beta _{+}) $ is the unique real number for which
\begin{gather*}
-u''(t)=\theta ( \alpha _{+}+\beta _{+})
u^{+}(t)+\lambda \alpha _{+}u^{-}(t)-\lambda \beta _{+}u^{+}(t),\ t\in
( \xi ,\eta ) , \\
au(\xi )-b\lim_{t\to \xi} p(t)u'(t)=0, \\
cu(\eta )+d\lim_{t\to \eta} p(t)u'(t)=0.
\end{gather*}
has a solution in $S_k^{\nu }$.

Assume that $k=2l$ with $l\geq 1$ and $\nu =+$ or $-$ (the other cases can
be checked similarly). We have from Corollary \ref{coro2} that
\begin{equation}
\begin{aligned}
&\Lambda _{a,b}( ( \theta ^{\ast }( \lambda ) -\lambda
) \beta _{+}+\theta ^{\ast }( \lambda ) \alpha _{+})
+\Lambda _{c,d}( ( \theta ^{\ast }( \lambda ) -\lambda
) \alpha _{+}+\theta ^{\ast }( \lambda ) \beta _{+})
\\
&+\frac{\pi ( l-1) }{\sqrt{( \theta ^{\ast }( \lambda
) -\lambda ) \beta _{+}+\theta ^{\ast }( \lambda )
\alpha _{+}}}+\frac{\pi ( l-1) }{\sqrt{( \theta ^{\ast
}( \lambda ) -\lambda ) \alpha _{+}+\theta ^{\ast }(
\lambda ) \beta _{+}}}=\eta -\xi .
\end{aligned}  \label{a*+}
\end{equation}
Taking into consideration the fact that $\Lambda _{a,b}$
and $\Lambda _{c,d}$ are decreasing functions, we obtain
from $( \ref{a*+}) $ that
\begin{align*}
\eta -\xi
&\leq \Lambda _{a,b}( \theta ^{\ast }( \lambda )
\alpha _{+}) +\Lambda _{c,d}( \theta ^{\ast }( \lambda
) \beta _{+}) +\frac{\pi ( l-1) }{\sqrt{\theta
^{\ast }( \lambda ) \alpha _{+}}}+\frac{\pi ( l-1) }{
\sqrt{\theta ^{\ast }( \lambda ) \beta _{+}}} \\
&\leq \pi l( \frac{1}{\sqrt{\theta ^{\ast }( \lambda )
\alpha _{+}}}+\frac{1}{\sqrt{\theta ^{\ast }( \lambda ) \beta
_{+}}})
\end{align*}
leading to
\[
\theta ^{\ast }( \lambda ) \leq \pi ^{2}l^{2}( \eta -\xi
) ^{2}( \frac{1}{\sqrt{\alpha _{+}}}+\frac{1}{\sqrt{\beta _{+}}}
) ^{2},
\]
which contradicts $\lim_{\lambda \to +\infty }\theta ^{\ast }(
\lambda ) =+\infty $.

Thus there exists $\lambda _k^{\nu }$ such that $\theta _k^{\nu }(
\lambda _k^{\nu }) =\lambda _k^{\nu }$ and $\lambda _k^{\nu }$
is a half-eigenvalue of \eqref{bvp2}. Uniqueness and simplicity of
 $\lambda _k^{\nu }$ follow from Lemma \ref{lemmex1}. Finally, the
monotonicity of $\lambda _k^{\nu }$ with respect of the weights $\alpha $
and $\beta $ follows directly from Proposition \ref{beres}.
\end{proof}

\begin{proposition}\label{proposition2}
Assume that $p\equiv 1$, $\alpha ,\beta $ are
nonnegative and continuous on $[ \xi ,\eta ] $ and the set $
\{ t\in [ \xi ,\eta ] :\alpha ( t) \beta (
t) >0\} $ has positive measure. Then the set of half-eigenvalues
of bvp \eqref{bvp2} consists of two increasing sequences of simple
half-eigenvalues $(\lambda _k^{+})_{k\geq 1}$ and $(\lambda
_k^{-})_{k\geq 1}$, such that for all $k\geq 1$ and $\nu =+$ or $-$, the
corresponding half-lines of solutions are in
$\{ \lambda _k^{\nu}\} \times S_k^{\nu }$.
Moreover for all $k\geq 1$ and $\nu =+$ or $-,\ \lambda _k^{\nu }$
is a decreasing function with respect to the
weights $\alpha $ and $\beta $ lying in $C[ \xi ,\eta ] $.
\end{proposition}

\begin{proof}
For $n\geq 1$, $\alpha _n=\alpha +\frac{1}{n}$ and
$\beta _n=\beta + \frac{1}{n}$,  let
$\lambda _{k,n}^{\nu }=\lambda _k^{\nu }( \alpha_n,\beta _n) $
be the half-eigenvalue given by Proposition \ref{thm8} associated
with the eigenvector $\phi _n\in \Theta _k^{\nu }$.
Because $( \alpha _n) $ and $( \beta _n) $ are
decreasing sequences, we have from Proposition \ref{thm8} that
 $(\lambda _{k,n}^{\nu }) _n$ is nondecreasing.
Now let $I_0=[\xi _0,\eta _0]\subset ( \xi ,\eta ) $ be such that
$\alpha \beta >0$ in $I_0$ and set
$\vartheta =\min ( \alpha ,\beta ) $. Hence, we have
$\vartheta _n=\min ( \alpha _n,\beta_n) =\vartheta +\frac{1}{n}\geq \vartheta $
and we deduce, from the montonicity property in Proposition \ref{thm8}
and Properties 5 and 6 in Theorem \ref{thm2}, that
\[
\lambda _{k,n}^{\nu }=\lambda _k^{\nu }( \alpha _n,\beta
_n) \leq \lambda _k^{\nu }( \vartheta _n,\vartheta
_n) =\mu _k( \vartheta _n,[ \xi ,\eta ] )
\leq \mu _k( \vartheta _n,I_0) \leq \mu _k(
\vartheta ,I_0) ,
\]
and the sequence $( \lambda _{k,n}^{\nu }) _n$ converges to
some $\lambda _k^{\nu }>0$, which is by Lemma \ref{lalimite} and Lemma
\ref{lemmex1}, a simple half-eigenvalue of bvp \eqref{bvp2} having an
eigenvector $\phi =\lim \phi _n\in \overline{\Theta _k^{\nu }}$ (up to a
subsequence). Because the functions $u\in \partial \Theta _k^{\nu }$ have
a double zero, Lemma \ref{zero} guarantees that $\phi \in \Theta _k^{\nu }$.

Let $\alpha _1$ be a nonnegative continuous function such that the set
$\{t\in [ \xi ,\eta ] :\alpha _1( t) \beta (t) >0\} $ has a positive measure
and $\alpha \leq \alpha _1$. We
have from Proposition \ref{thm8} that
\[
\lambda _k^{\nu }\Big( \alpha +\frac{1}{n},\beta +\frac{1}{n}\Big) \leq
\lambda _k^{\nu }\Big( \alpha _1+\frac{1}{n},\beta +\frac{1}{n}\Big) .
\]
Letting $n\to \infty $ we obtain
$\lambda _k^{\nu }( \alpha,\beta ) \leq \lambda _k^{\nu }( \alpha _1,\beta ) $.
Similarly we prove that $\lambda _k^{\nu }$ is nonincreasing with respect
to the weight $\beta$. The proof is complete.
\end{proof}

\subsection{Proof of Theorem \ref{thm10}}

Let $\varphi $ and $\rho _0$ be as in Section 2 and note that $\lambda $
is a half-eigenvalue with an eigenvector $u$ of \eqref{bvp2} if and only if
$\lambda /\rho _0$ is a half-eigenvalue with the eigenvector
$v=u\circ\varphi ^{-1}$ of the bvp
\begin{equation}
\begin{gathered}
-v''(t)=\frac{\lambda }{\rho _0}\Big( \widetilde{
\alpha }( t) v^{+}(t)-\widetilde{\beta }( t)
v^{-}(t)\Big) ,\quad t\in ( \xi ,\eta ) , \\
av(\xi )-b\rho _0v'(\xi )=0, \\
cv(\eta )+d\rho _0v'(\eta )=0,
\end{gathered}  \label{HECaux}
\end{equation}
where
\[
\widetilde{\alpha }( t) =p( \varphi ^{-1}(t) ) \alpha ( \varphi ^{-1}( t) ),\quad
\widetilde{\beta }( t) =p( \varphi ^{-1}( t) ) \beta ( \varphi ^{-1}( t))
\]
are integrable functions. So, it suffices to prove Theorem \ref{thm10} with
$p\equiv 1$. To this aim, let $( \alpha _n) $ and
$( \beta_n) $ be two sequences in $C_{c}( \xi ,\eta ) $ such that
$\lim \alpha _n=\alpha $ and $\lim \beta _n=\beta $ in
$L^{1}( \xi,\eta ) $, and let
$\lambda _{k,n}^{\nu }=\lambda _k^{\nu }(\alpha _n,\beta _n) $
be the half-eigenvalue given by Proposition
\ref{proposition2} associated with an eigenvector $\phi _n$.
Let $\vartheta _n=\inf ( \alpha _n,\beta _n)$,
$\theta _n=\sup( \alpha _n,\beta _n) $, and
$\theta =\sup ( \alpha,\beta ) \geq \vartheta =\inf ( \alpha ,\beta ) >0$
on some interval $I_0=[ \xi _0,\eta _0] \subset ( \xi ,\eta) $.
We have that $\lim \vartheta _n=\vartheta $ in
$L^{1}( \xi,\eta ) $. Then, we deduce from the monotonicity property in
Proposition \ref{proposition2} and Property 5 in Theorem \ref{Preli4} that
\begin{align*}
0 &<\mu _k( \vartheta ) -\epsilon
=\mu _k( \vartheta_n) \\
&=\lambda _k^{\nu }( \theta _n,\theta _n)
\leq \lambda _{k,n}^{\nu } \\
&= \lambda _k^{\nu }( \alpha _n,\beta _n)
\leq \lambda _k^{\nu }( \vartheta _n,\vartheta _n) \\
&=\mu _k(\vartheta _n) \leq \mu _k( \vartheta ) +C
\end{align*}
where the constant $\epsilon $ and $C$ are respectively small enough and
large enough. Let $\lambda _{k,s}^{\nu }=\lim \sup \lambda _{k,n}^{\nu }$
and $\lambda _{k,i}^{\nu }=\lim \inf \lambda _{k,n}^{\nu }$. We have from
Lemma \ref{lalimite} that $\lambda _{k,s}^{\nu }$ and $\lambda _{k,i}^{\nu }$
are half-eigenvalues of \eqref{bvp2}. Then we deduce from Lemma \ref{lemmex1}
that $\lambda _k^{\nu }=\lambda _{k,s}^{\nu }=\lambda _{k,i}^{\nu }$ is
the unique and simple half-eigenvalue of \eqref{bvp2}. The same arguments as
those used in the proof of Proposition \ref{proposition2} show that the
eigenvector associated with $\lambda _k^{\nu }$ belongs to $S_k^{\nu}\cap Y$.

\subsection{Proof of Theorem \ref{thm1}}

\subsubsection{Proof of uniqueness of $\lambda _1^{\nu}$}\label{gfd}

\begin{lemma}\label{unicite1}
Assume that Hypotheses \eqref{hyp1} and \eqref{hyp2} hold
and $\alpha ,\beta \in K_{G}$. Then for $\nu =+$ or $-$,
bvp \eqref{bvp2} admits at most one half-eigenvalue having an eigenvector
in $S_1^{\nu }$.
\end{lemma}

\begin{proof}
Suppose that $\lambda _1^{+}$ is a half-eigenvalue having an eigenvector
$\phi _1\in S_1^{+}$ (uniqueness of $\lambda _1^{-}$ can be proved in
the same way), then $1/\lambda _1^{+}$ is a positive eigenvalue of the
positive operator $L_{\alpha }:E\to E$ defined by
\[
L_{\alpha }u(t)=\int_{\xi }^{\eta }G( t,s) \alpha (s)u(s) ds.
\]
So, we have that $r( L_{\alpha }) >0$ and since the cone of
nonnegative functions is total in $E$, $r( L_{\alpha }) $ is a
positive eigenvalue of $L_{\alpha }$ and
\begin{equation}
\lambda _1^{+}\geq 1/r( L_{\alpha }) .  \label{M0}
\end{equation}

Let $( \xi _n) $ and $( \eta _n) $ be two
sequences in $( \xi ,\eta ) $ such that
$\lim \xi _n=\xi$, $\lim \eta _n=\eta$,
$( \xi _n)$ is decreasing,
$( \eta_n)$ is increasing,
and set
\[
\alpha _n(t)=\begin{cases}
\inf ( \alpha (t),\alpha ( \xi _n) ) ,& \text{if }t\leq \xi _n, \\
\alpha (t), & \text{if  }t\in ( \xi _n,\eta _n) ,\\
\inf ( \alpha (t),\alpha ( \eta _n) ) ,& \text{if } t\geq \eta _n,
\end{cases}
\]
and let $L_n:E\to E$ be the linear operator defined by
\begin{equation}
L_nu(t)=\int_{\xi }^{\eta }G(t,s)\alpha _n(s)u(s)ds.  \label{Ln}
\end{equation}
We see that for all $n\in\mathbb{N}$, $L_n\leq L_{\alpha }$.
Then from Lemma \ref{lemmeXX} we have  $r(L_n)\leq r(L_{\alpha })$.

We have that for all $n\in\mathbb{N}$, $\lambda _1^{n}=1/r( L_n) >0$
is the unique positive eigenvalue associated with a positive eigenvector
$\phi _1^{n}$ to the linear bvp
\begin{gather*}
-( pu') '(t)=\lambda \alpha _n(t)u(t),\quad \text{a.e. } t\in ( \xi ,\eta ) , \\
au(\xi )-b\lim_{t\to \xi}p(t)u'(t)=0, \\
cu(\eta )+d\lim_{t\to \eta} p(t)u'(t)=0.
\end{gather*}
Moreover, $\alpha _n\to \alpha $ in
$L_{G}^{1}[ \xi ,\eta ] $. So, we have from Lemmas \ref{regularite}
and \ref{radius} that
\begin{equation}
\lim \lambda _1^{n}=1/r( L_{\alpha }) \leq \lambda _1^{+}.
\label{M4}
\end{equation}
Before proving uniqueness, note that, if $\lambda $ is a positive eigenvalue
of bvp \eqref{bvp3} associated with an eigenvector $\phi $, then there
exists a subinterval
$[ \gamma ,\delta ] \subset ( \xi ,\eta) $ such that $\alpha ( t) \phi ( t) >0$ for
almost all $t\in [ \gamma ,\delta ]$. Indeed if this does not
occur, we obtain the contradiction
\[
\phi (t)=\lambda \int_{\xi }^{\eta }G(t,s)\alpha (s)\phi (s)ds=0\quad
\text{for all }t\in ( \xi ,\eta ) .
\]
This means also that
\[
\phi (t)=\lambda \int_{\xi }^{\eta }G(t,s)\alpha (s)\phi (s)ds>0\text{ }
\text{for all } t\in ( \xi ,\eta ) .
\]
Set
\[
\psi _n=L_n\phi _1\leq L_{\alpha }\phi _1=( \lambda _1^{+}) ^{-1}\phi _1.
\]
Observe that $\psi _n$ satisfies
\begin{equation}
\begin{gathered}
-( p'\psi _n) '(t)=\alpha _n(t)\phi
_1(t)\geq \lambda _1^{+}\alpha _n(t)\psi _n( t) \quad
\text{a.e. } t\in ( \xi ,\eta ) , \\
a\psi _n(\xi )-b\lim_{t\to \xi}p(t)\psi _n'(t)=0, \\
c\psi _n(\eta )+d\lim_{t\to \eta} p(t)\psi_n'(t)=0.
\end{gathered}  \label{jhg}
\end{equation}
Multiplying the differential inequality in \eqref{jhg} by
$\phi _1^{n}$ (the eigenvector of $\lambda _1^{n}$) and integrating over
$[ \xi ,\eta ] $ we obtain
\[
\int_{\xi }^{\eta }-( p\psi _n') '\phi_1^{n}
\geq \lambda _1^{+}\int_{\xi }^{\eta }\alpha _n\psi _n\phi_1^{n}.
\]
We find, after two integration by parts of the left hand side,
\[
\lambda _1^{n}\int_{\xi }^{\eta }\alpha _n\psi _n\phi _1^{n}\geq
\lambda _1^{+}\int_{\xi }^{\eta }\alpha _n\psi _n\phi _1^{n},
\]
leading to
$\lambda _1^{+}\leq \lambda _1^{n}$ for all $n\geq 1$,
from which we have
\begin{equation}
\lambda _1^{+}\leq \lim \lambda _1^{n}.  \label{M5}
\end{equation}
At the end, combining \eqref{M5} with \eqref{M4}, we obtain
$\lambda _1^{+}=1/r(L_{\alpha })$, that is $1/r(L_{\alpha })$ is the unique half
eigenvalue of \eqref{bvp2} having an eigenvector in $S_1^{+}$.
\end{proof}

\subsubsection{Proof of uniqueness of $\lambda _k^{\nu }$, $k\geq 2$}

\begin{lemma} \label{unicite2}
Assume that Hypotheses \eqref{hyp1} and \eqref{hyp2} hold
and $\alpha ,\beta \in K_{G}$. Then for each integer $k\geq 1$ and $\nu =+$
or $-$, bvp \eqref{bvp2} admits at most one half-eigenvalue having an
eigenvector in $S_k^{\nu }$.
\end{lemma}

\begin{proof}
To the contrary, assume that $\lambda _1$ and $\lambda _2$ are two
half-eigenvalues having, respectively, the eigenvectors
$\phi _1,\phi _2\in S_k^{\nu }$ with the sequences of simple zeros
$(x_i) _{1\leq i\leq k}$ and $( y_i) _{1\leq i\leq k}$.
In the spirit of Theorem \ref{Preli1}, assume that $x_1\leq y_1$ and let
$i_0,j_0\in \{ 1,\ldots ,k\} $ such that
$x_{i_0}\leq z_{j_0}\leq z_{j_0+1}\leq x_{i_0+1}$, and without loss
 of generality,
suppose that $\phi _1\geq 0$ and $\phi _2\geq 0$ in each of the
intervals $[ \xi ,x_1] $ and $[ z_{j_0},z_{j_0+1}] $.
Let $( \xi _n) $ and $( \eta _n) $ be
the sequences given in the proof of Lemma \ref{unicite1} and set
\[
\alpha _n^{1}(t)=\begin{cases}
\inf ( \alpha (t),\alpha ( \xi _n) ) ,& \text{if }t\leq \xi _n, \\
\alpha (t),& \text{if }t\in ( \xi _n,z_1) .
\end{cases}
\]
From Lemma \ref{unicite1}, we have
$\lambda _1=\lim \mu _1( \alpha _n^{1},[ \xi ,x_1] ) $ and
$\lambda _2=\lim \mu _1( \alpha _n^{1},[ \xi ,z_1] ) $, and from
Property 4 of Theorem \ref{Preli4}, that for all
$n\geq 1$, $\mu _1(\alpha _n^{1},[ \xi ,x_1] ) \geq \mu _1( \alpha
_n^{1},[ \xi ,z_1] ) $. Letting $n\to \infty $
we obtain $\lambda _1\geq \lambda _2$.

Now we will discuss the cases $z_{j_0+1}<\eta $ and $z_{j_0+1}=\eta $.
If $z_{j_0+1}<\eta$, then integrating on
$[ z_{j_0},z_{j_0+1} ]$, we obtain
\[
0\geq \int_{z_{j_0}}^{z_{j_0+1}}-( p\phi _1')
'\phi _2+( p\phi _2') '\phi_1
=( \lambda _1-\lambda _2)
\int_{z_{j_0}}^{z_{j_0+1}}\alpha \phi _1\phi _2,
\]
leading to $\lambda _1=\lambda _2$.

If $z_{j_0+1}=\eta $, then considering
\[
\alpha _n^{2}(t)=\begin{cases}
\alpha (t), & \text{if }t\in ( x_{i_0},\eta _n) , \\
\inf ( \alpha (t),\alpha ( \nu _n) ) ,&
\text{if }t\geq \eta _n,
\end{cases}
\]
we have that $\lambda _1=\lim \mu _1( \alpha _n^{2},[x_{i_0},\eta ] ) $ and
$\lambda _2=\lim \mu _1(\alpha _n^{2},[ z_{j_0},\eta ] ) $,
and from Property 3 of Theorem 2.5, that for all
$n\geq 1$, $\mu _1( \alpha _n^{2},[z_{j_0},\eta ] ) \geq \mu _1( \alpha _n^{2},[
x_{i_0},\eta ] ) $. So, letting $n\to \infty $ we obtain
also in this case $\lambda _1=\lambda _2$.
 This completes the proof.
\end{proof}

\subsubsection{Proof of existence of $(\lambda _k^{\nu })_{k\geq 1}$} \label{sss}

Let $( \xi _n) $ and $( \eta _n) $ be the
sequences introduced in the proof of Lemma \ref{unicite1} and consider
\begin{gather*}
\alpha _n(t)=\begin{cases}
\alpha (t), &\text{if } t\in ( \xi _n,\eta _n) , \\
0, & \text{if }t\notin ( \xi _n,\eta _n) ,
\end{cases}
\\
\beta _n(t)=\begin{cases}
\beta (t), &\text{if }t\in ( \xi _n,\eta _n) ,\\
0,& \text{if }t\notin ( \xi _n,\eta _n) .
\end{cases}
\end{gather*}

For all $n,k\geq 1$ and $\nu =+$ or $-$, let $\lambda _{k,n}^{\nu }$ be the
unique half-eigenvalue of
\begin{gather*}
-( pu') '(t)=\lambda ( \alpha _n(
t) u^{+}(t)-\beta _n( t) u^{-}(t)) \quad \text{a.e. }
t\in ( \xi ,\eta ) , \\
au(\xi )-b\lim_{t\to\xi} p(t)u'(t)=0, \\
cu(\eta )+d\lim_{t\to\eta} p(t)u'(t)=0,
\end{gather*}
having  eigenvector $\phi _n\in S_k^{\nu }$ with
$\| \phi _n\| =1$, $\theta _n=\sup ( \alpha _n,\beta _n) $
 and $\theta =\sup ( \alpha ,\beta ) >0$ in some closed interval
$I_0\subset ( \xi ,\eta )$. Since $( \alpha _n) $
and $( \beta _n) $ are nondecreasing sequences, we have from
Property 2 in Theorem \ref{thm10} that, for all $n\geq 1$,
\[
\lambda _{k,n}^{\nu }\geq \lambda _{k,n+1}^{\nu },
\]
and
\begin{equation}
\lambda _{k,n}^{\nu }=\lambda _k^{\nu }( \alpha _n,\beta
_n) \geq \lambda _k^{\nu }( \theta _n,\theta _n)
=\mu _k( \theta _n) >\mu _1( \theta _n) .
\label{fou}
\end{equation}
Because $\mu _1( \theta _n) =1/r( L_{\theta_n})$,
$\mu _1( \theta ) =1/r( L_{\theta }) $
and $L_{\theta _n}\to L_{\theta }$ in operator norm, it follows
from Lemma \ref{radius} that, for $\epsilon >0$ small enough,
\[
\lambda _{k,n}^{\nu }\geq \lambda _{k,n+1}^{\nu }\geq \mu _1( \theta
) -\epsilon >0.
\]
Thus,  from Lemma \ref{lalimite} we have that
$\lambda _k^{\nu}=\lim_{n\to \infty }\lambda _{k,n}^{\nu }$
is a half-eigenvalue of \eqref{bvp2}) having an eigenvector $\psi $
 (as it it shown in proof of Lemma \ref{lalimite} $\psi =\lim \phi _n$).

In view of Lemma \ref{zero}, it remains to show that $\psi \in S_k^{\nu }$.
To the contrary, assume that $\psi \in S_{l}^{+}$ with $l\neq k$ and let
 $( z_{j}) _{j=1}^{j=l-1}$ be the sequence of interior zeros of
$\psi $ and $[ \xi _1,\eta _1] \subset ( \xi ,\eta) $ such that
\[
\xi _1<z_1<z_2<\cdots <z_{l-1}<\eta _1.
\]
Choose $\delta >0$ small enough and set
$I_{j}=( z_{j}-\delta,z_{j}+\delta ) $ for
$j\in \{ 1,\ldots ,l-1\} $. There exists
$n_{\ast }\in\mathbb{N}$ such that for all $n\geq n_{\ast }$,
$\phi _n\psi >0$ in all the intervals
$[ \xi _1,z_1-\delta ] $, $[ z_{k-1}+\delta ,\eta _1]$,
$[ z_{j}+\delta ,z_{j+1}-\delta ]$,
$j\in\{ 1,\ldots ,l-2\} $.

Fix $j\in \{ 1,\ldots ,l-1\} $. There exists $n_{j}\geq n_{\ast }$
such that the function $\phi _n$ has exactly one zero in $I_{j}$.
Otherwise if there is a subsequence $(\phi _{n_i})$ such that for all
$ i\geq 1$, $\phi_{n_i}$ has at least two zeros, then we can choose
$x_{n_i}^{1}$ and $x_{n_i}^{2}$ in $I_{j}$ such that
\[
\phi _{n_i}^{[ 1] }( x_{n_i}^{1}) \leq 0\leq \phi
_{n_i}^{[ 1] }( x_{n_i}^{2}) .
\]
Let
\begin{gather*}
x_{\inf }^{1}=\lim \inf x_{n_i}^{1}, \quad x_{\sup }^{1}=\lim \sup x_{n_i}^{1}
\\
x_{\inf }^{2}=\lim \inf x_{n_i}^{1},\quad  x_{\sup }^{2}=\lim \inf
x_{n_i}^{1}.
\end{gather*}
Hence,  since $\psi =\lim \phi _n/\| \phi _n\| $ we have
\[
\psi ( x_{\inf }^{1}) =\psi ( x_{\inf }^{2}) =\psi
( x_{\sup }^{1}) =\psi ( x_{\sup }^{2}) =0
\]
leading to
\[
\lim x_{n_i}^{1}=\lim x_{n_i}^{2}=z_{j}.
\]
Moreover,  from Remark \ref{regularite1} it follows that
\[
\psi ^{[ 1] }( z_{j}) =\lim \phi _{n_{l}}^{[ 1
] }( x_{n_i}^{1}) =\lim \phi _{n_{l}}^{[ 1]
}( x_{n_i}^{2}) =0,
\]
contradicting the simplicity of $z_{j}$.

Now, we claim that there exists $n^{\ast }\in\mathbb{N}$ such that for all
$n\geq n^{\ast },\ \phi _n$ does not vanish in the
intervals $( \xi ,\xi _1) $ and $( \eta _1,\eta )$.
 Again, to the contrary, assume that there is a subsequence
$(\phi_{n_i}) $ such that for all $i\geq 1$, $\phi _{n_i}$ has at least one
zero. Let $x_{n_i}\in ( \xi ,\xi _1) $ be the first zero of
$\phi _{n_i}$. In this case, we have that
\[
\lim x_{n_i}=\xi,\quad \psi ( \xi ) =\phi _{n_i}(\xi ) =0.
\]
Moreover, for all $i\geq 1$, $\phi _{n_i}$ satisfies
\begin{equation}
\begin{gathered}
-( p\phi _{n_i}') '(t)=\mu _{n_i}\alpha
_{n_i}(t)\phi _{n_i}( t) \quad \text{a.e. } t\in ( \xi,x_{n_i}) , \\
\phi _{n_i}(\xi )=\phi _{n_i}(x_{n_i})=0 \\
\phi _{n_i}>0\ in\ ( \xi ,x_{n_i}) .
\end{gathered} \label{ere10}
\end{equation}
Clearly, Equation \eqref{ere10} implies that
$\mu _{n_i}=\mu _1( \alpha _{n_i},[ \xi ,x_{n_i}] ) $. Taking
into consideration $\lim \alpha _n=\alpha $ in
$L_{G}^{1}[ \xi ,\eta ] $ and
$\mu _{n_i}=\mu _1( \alpha _{n_i},[ \xi,x_{n_i}] ) =1/r( L_{\alpha }^{n_i}) $ where
$L_{\alpha }^{n_i}:C[ \xi ,x_{n_i}] \to C[ \xi,x_{n_i}] $ is defined by
\[
L_{\alpha }^{n_i}u( t)
=\int_{\xi }^{x_{n_i}}G_{n_i}(t,s) \alpha ( s) u( s) ds,
\]
from Lemma \ref{radius} we obtain
\begin{equation}
\mu _{n_i}=\mu _1( \alpha _{n_i},[ \xi ,x_{n_i}]
) =\frac{1}{r( L_{\alpha _{n_i}}^{n_i}) }\geq \frac{1}{
r( L_{\alpha }^{n_i}) }=\mu _1( \alpha ,[ \xi
,x_{n_i}] ) .  \label{XNC1}
\end{equation}
Thus, combining Lemma \ref{lemmex} with \eqref{XNC1}), we
obtain the contradiction
\[
\widetilde{\lambda }=\lim \mu _{n_i}\geq \lim \mu _1( \alpha ,
[ \xi ,x_{n_i}] ) =\lim \frac{1}{r( L_{\alpha
}^{n_i}) }=+\infty .
\]

Hence, we conclude that for all
$n\geq n_{\infty }=\max \{ n_{\ast},n^{\ast },n_1,\ldots ,n_{k-1}\} $,
 $\phi _n$ has exactly $(l-1) $ simple zeros in $( \xi ,\eta ) $
contradicting $\phi _n\in S_k^{+}$.

Finally, letting $n\to \infty $ in $\lambda _{k,n}^{\nu }<\lambda
_{k+1,n}^{\nu }$ we obtain $\lambda _k^{\nu }\leq \lambda _{k+1}^{\nu }$.

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

The existence of $( \mu _k( q) ) _{k\geq 1}$ as
a nondecreasing sequence follows from Theorem \ref{thm1} when taking
$\alpha =\beta =q$ in bvp \eqref{bvp2} and for all
$k\geq 1$, $\mu _k(q) $ has an eigenvector $\phi _k\in S_k$.
We have from Lemma \ref{lemmex1} and assertion 2 in Lemma \ref{lemmeAB}
that $\mu _k(q) $ is simple for all $k\geq 3$ and Assertion 1 is proved.
Assertion 2 follows from the monotonicity of the sequence
$( \mu _k(q) ) _{k\geq 1}$ and the simplicity of $\mu _k( q) $
for $k\geq 3$. Assertion 3 follows from Lemma \ref{lemmex1}, and assertion 1
in Lemma \ref{lemmeAB} and Lemma \ref{unicite1}. Assertion 4 is obvious.

Assertion 5 follows from the monotonicity property of half-eigenvalues in
Theorem \ref{thm1} and Assertion 6 is obtained when letting
$n\to \infty $ in the relation
\[
\mu _k( q_n,[ \xi ,\eta ] ) \leq \mu _k(q_n,[ \xi _1,\eta _1] ) ,
\]
where
\[
q_n(t)=\begin{cases}
q(t), & \text{if } t\in ( \xi _n,\eta _n) , \\
0, &\text{if } t\notin ( \xi _n,\eta _n) ,
\end{cases}
\]
and $( \xi _n) _{n\geq 1},( \eta _n) _{n\geq 1}$
are those in the proof of Lemma \ref{unicite1}.

It remains to prove Assertion 7. Let $( q_n) \subset K_{G}$ be
a sequence converging to $q\in K_{G}$ in $L_{G}^{1}[ \xi ,\eta ] $,
and $[ \xi _0,\eta _0] \subset ( \xi ,\eta ) $
such that $q>0$ in $[ \xi _0,\eta _0] $. We have then from
Property 6 and Property 5 in Theorem \ref{Preli4}
\[
\mu _k( q_n,[ \xi ,\eta ] ) \leq \mu _k(
q_n,[ \xi _0,\eta _0] ) \leq \mu _k( q,[
\xi _0,\eta _0] ) +C.
\]
Set $\mu ^{1}=\lim \inf \mu _k( q_n,[ \xi ,\eta ]) $ and
$\mu ^{2}=\lim \sup \mu _k( q_n,[ \xi ,\eta] ) $.
There exist two subsequences $( \mu _k(q_{n_i},[ \xi ,\eta ] ) ) $ and
$( \mu_k( q_{n_{j}},[ \xi ,\eta ] ) ) $ of $(\mu _k( q_n,[ \xi ,\eta ] ) ) $
converging respectively to $\mu ^{1}$ and $\mu ^{2}$.
Applying Lemma \ref{lalimite}, we obtain that $\mu ^{1}$ and $\mu ^{2}$ are
eigenvalues of \eqref{bvp3}.
Furthermore, arguing as in Subsection 4.3.3, we see that the eigenvectors
associated with $\mu ^{1}$ and $\mu ^{2}$ belongs to $S_k$.
Thus, we deduce from Lemmas \ref{unicite1} and \ref{unicite2} that
\[
\mu ^{1}=\mu ^{2}=\lim \mu _k( q_n,[ \xi ,\eta ] )
=\mu _k( q,[ \xi ,\eta ] ) .
\]
This completes the proof

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

Consider the bifurcation bvp associated with bvp \eqref{bvp1},
\begin{equation}
\begin{gathered}
-( pu') '(t)=\mu q_0(t)u( t) +\mu
g(t,u(t)),\quad \text{a.e. }t\in ( \xi ,\eta ) , \\
au(\xi )-b\lim_{t\to\xi} p(t)u'(t)=0, \\
cu(\eta )+d\lim_{t\to\eta} p(t)u'(t)=0,
\end{gathered}  \label{bvp400}
\end{equation}
where $\mu $ is a real parameter and
$g(t,u)=f( t,u) -q_0( t) u$ and in all that follows, we denote by
$( \mu _k(q_0) ) _{k\geq 1}$ the sequence of eigenvalues obtained from
Theorem \ref{thm2} for the bvp
\begin{gather*}
\pounds u(t)=\mu q_0(t)u(t),\quad \text{a.e.  }t\in (\xi ,\eta ),
\\
au(\xi )-b\lim_{t\to \xi }p( t) u'(t)=0, \\
cu(\eta )+d\lim_{t\to \eta }p( t) u'(t)=0,
\end{gather*}
and by $( \chi _k^{\nu }) _{k\geq 1}$, with $\nu =+$ or $-$,
the two sequences of half-eigenvalues of the bvp
\begin{gather*}
-( pu') '(t)=\chi ( \alpha _{\infty
}(t)u^{+}( t) -\beta _{\infty }(t)u^{-}( t) ) ,
\quad \text{a.e.  }t\in ( \xi ,\eta ) , \\
au(\xi )-b\lim_{t\to\xi} p(t)u'(t)=0, \\
cu(\eta )+d\lim_{t\to\eta} p(t)u'(t)=0,
\end{gather*}
given by Theorem \ref{thm1}.

Applying $\pounds ^{-1}$, we obtain that bvp \eqref{bvp400} is equivalent to
the equation
\[
u=\mu L_{q_0}u+\mu H( u)
\]
where $H:E\to E$ is defined by
\[
Hu(s)=\int_{\xi }^{\eta }G( t,s) g(s,u( s) )ds
\]
and is completely continuous.

\begin{lemma} \label{4.1}
Assume that \eqref{hyp1}-\eqref{hyp3} hold. Then from each
$\mu _k( q_0)$, with $k\geq 3$, bifurcate two unbounded components
(in $\mathbb{R}\times E$), $\Gamma _k^{+}$ and $\Gamma _k^{-}$ such
that for all $k\geq 3$ and $\nu =+$ or $-$,
$\Gamma _k^{\nu }\subset\mathbb{R}\times S_k^{\nu }$.
\end{lemma}

\begin{proof}
First, note that Hypothesis \eqref{hyp3} implies that
$H( u)=\circ ( \| u\| _{\infty }) $ near $0$. Indeed,
we have for $( u_n) \subset E$ with
$\lim \|u_n\| _{\infty }=0$
\[
\frac{| Hu_n( t) | }{\|
u_n\| _{\infty }}\leq \int_{\xi }^{\eta }R_n( s) ds\quad
\text{where }
 R_n( s) =G( s,s) \frac{|
f( s,u_n( s) ) -q_0( s) u_n(
s) | }{\| u_n\| _{\infty }}.
\]
Then, it follows from \eqref{hyp4} that
\begin{align*}
R_n( s)
&\leq G( s,s) ( \gamma _{\infty
}( s) +\delta _{\infty }( s) +q_0( s)
) \frac{| u_n( s) | }{\|u_n\| _{\infty }} \\
&\leq G( s,s) ( \gamma _{\infty }( s) +\delta
_{\infty }( s) +q_0( s) ) \in L^{1}[ \xi,\eta ]
\end{align*}
and from \eqref{hyp3} that
\begin{align*}
R_n( s)
&\leq G( s,s) | \frac{f(s,u_n( s) ) }{u_n( s) }-q_0( s)
| \frac{| u_n( s) | }{\|
u_n\| _{\infty }} \\
&\leq G( s,s) | \frac{f( s,u_n( s)) }{u_n( s) }-q_0( s) |
\to 0
\end{align*}
as $n\to +\infty$, $s\in [ \xi ,\eta]$ a.e..


So, by the Lebesgue dominated convergence theorem, we have
\[
\lim \frac{H(u_n)}{\| u_n\| _{\infty }}=0;
\]
that is, $H(u)=o( \| u\| _{\infty })$ at $0$.

Since for all $k\geq 3$, $\mu _k( q_0) $ is of algebraic
multiplicity one,  from \cite[Theorem 1.40]{rab12}  we conclude that from
each $( \mu _k( q_0) ,0) $ with $k\geq 3$,
bifurcate two components $\Gamma _k^{1}$ and $\Gamma _k^{2}$ of
nontrivial solutions of bvp \eqref{bvp400} such that for $i=1,2$,
 $\Gamma_k^{i}$ is either unbounded in $\mathbb{R}\times E$ or meets
$( \mu _{l}( q_0) ,0) $ where $l\neq k$.

Now, note that if $( \lambda ,u) \in \Gamma _k^{i}$ $i=1,2$ then
all zeros of $u$ are simple. This is due to the fact that
$( \lambda,u) $ satisfies also the bvp
\begin{gather*}
-( pv') '(t)=\lambda q_{u}( t)
v(t),\ t\in ( \xi ,\eta ) , \\
av(\xi )-b\lim_{t\to\xi} p(t)v'(t)=0, \\
cv(\eta )+d\lim_{t\to\eta} p(t)v'(t)=0,
\end{gather*}
with
\[
q_{u}( t) =\frac{f( t,u) }{u}.
\]
Since Hypothesis \eqref{hyp4} guarantees that
$q_{u}\in L_{G}^{1}[ \xi ,\eta ] $, we deduce from Theorem \ref{thm2}
that there exists an integer $j\geq 1$ such that $u\in S_{j}$.

Also, we claim that for all $k\geq 3$ and $i=1,2$, there exists a
neighborhood $V_k^{i}$ of $( \mu _k( q_0) ,0) $
such that $\Gamma _k^{i}\cap V_k^{1}\subset \mathbb{R}\times S_k$.
Let $( \mu _n,u_n) _{n\geq 1}\subset \Gamma _k^{i}$ be a sequence
converging to $( \mu _k,0) $. Thus,
 $v_n=u_n/\| u_n\| _{\infty }$ satisfies
\[
v_n=\mu _nL_{q_0}( v_n) +\mu _n\frac{H(
u_n) }{\| u_n\| _{\infty }}\quad \text{and}\quad
\| v_n\| _{\infty }=1.
\]
Since $L_{q_0}$ is compact and $H( u) =o( \|u\| _{\infty }) $ near $0$,
there exists a subsequence $( v_{n_{j}}) $ of $( v_n) $ converging to $v$ in $E$
satisfying
\[
v=\mu _k( q_0) L_{q_0}v\quad \text{and}\quad \|v\| _{\infty }=1.
\]
So, from Theorem \ref{thm2} we have  $v\in S_k$.

Let $( z_{j}) _{j=1}^{j=k-1}$ be the sequence of interior zeros
of $v$ and $[ \xi _1,\eta _1] \subset ( \xi ,\eta) $ such that
\[
\xi _1<z_1<z_2<\dots <z_{k-1}<\eta _1.
\]
Choose $\delta >0$ small enough and set
$I_{j}=( z_{j}-\delta ,z_{j}+\delta ) $ for $j\in \{ 1,\ldots ,k-1\} $. There
exists $n_{\ast }\in\mathbb{N}$ such that for all $n_{j}\geq n_{\ast }$,
$v_{n_{j}}v>0$ in all the intervals
$[ z_{j}+\delta ,z_{j+1}-\delta ]$  $j\in \{1,\ldots ,k-2\}$,
$[ \xi _1,z_1-\delta ] $, $[z_{k-1}+\delta ,\eta _1] $.

Fix $j\in \{ 1,\ldots ,k-1\} $. There exists $n_{j}\geq n_{\ast }$
such that the function $v_n$ has exactly one zero in $I_{j}$. Otherwise if
there is a subsequence $(v_{n_{l}})$ such that for all $l\geq 1$, $v_{n_{l}}$
has at least two zeros, then we can choose $x_{n_{l}}^{1}$ and
$x_{n_{l}}^{2} $ in $I_{j}$ such that
\[
v_{n_{l}}^{[ 1] }( x_{n_{l}}^{1}) \leq 0\leq
v_{n_{l}}^{[ 1] }( x_{n_{l}}^{2}) .
\]
Hence, we obtain
\[
\lim x_{n_{l}}^{1}=\lim x_{n_{l}}^{2}=z_{j},
\]
and from Lemma \ref{regularite} that
\[
v^{[ 1] }( z_{j}) =\lim v_{n_{l}}^{[ 1]
}( x_{n_{l}}^{1}) =\lim v_{n_{l}}^{[ 1] }(
x_{n_{l}}^{2}) =0,
\]
contradicting the simplicity of $z_{j}$.

Now, we claim that there exists $n^{\ast }\in\mathbb{N}$ such that for all
 $n\geq n^{\ast },\ v_n$ does not vanish in the
intervals $( \xi ,\xi _1) $ and $( \eta _1,\eta )$.
 To the contrary, assume that there is a subsequence $(v_{n_{l}})$ such
that, for all $l\geq 1$, $v_{n_{l}}$ has at least one zero.\ Let $
x_{n_{l}}\in ( \xi ,\xi _1) $ be the first zero of $v_{n_{l}}$.
In this case, we have that
\[
\lim x_{n_{l}}=\xi , \quad v( \xi ) =v_n( \xi) =0.
\]
Moreover, for all $l\geq 1$, $u_{n_{l}}$ satisfies
\begin{equation}
\begin{gathered}
-( pu_{n_{l}}') '(t)=\mu
_{n_{l}}q_{n_{l}}(t)u_{n_{l}}( t) ,\quad t\in ( \xi ,x_{n_{l}}) , \\
u_{n_{l}}(\xi )=u_{n_{l}}(x_{n_{l}})=0, \\
u_{n_{l}}>0\quad\text{in } ( \xi ,x_{n_{l}}) ,
\end{gathered}\label{ere1}
\end{equation}
where $q_{n_{l}}(t)=( f( t,u_{n_{l}}( t) )/u_{n_{l}}( t) ) $.
Clearly, Equation \eqref{ere1} implies that
 $\mu _{n_{l}}=\mu _1(q_{n_{l}}, [ \xi ,x_{n_{l}}] )$.
Taking into consideration Hypothesis \eqref{hyp4},
from Property 3 in Theorem \ref{thm2} we obtain that
\[
\mu _{n_{l}}=\mu _1( q_{n_{l}},[ \xi ,x_{n_{l}}] )
\geq \mu _1( \gamma _{\infty },[ \xi ,x_{n_{l}}] ) .
\]

So, we obtain as in Subsection \ref{sss} the contradiction
\[
\mu _k( q_0) =\lim \mu _{n_{l}}\geq \lim \mu _1(
\gamma _{\infty },[ \xi ,x_{n_{l}}] ) =+\infty .
\]

At this stage we conclude that, for all
$n\geq n_{\infty }=\max \{n_{\ast },n^{\ast },n_1,\ldots ,n_{k-1}\} $,
 $v_n$ has exactly $( k-1) $ simple zeros in $( \xi ,\eta ) $ and so the
existence of the neighborhood $V_k^{\nu }$.

Using the same arguments as those used above, we see that for all
$(\mu _0,u_0) \in \Gamma _k^{i}$ with $u_0\in S_{k_1}$, there
exists a neighborhood $W_0$ of $( \mu _0,u_0) $ such that
$W_0\cap \Gamma _k^{\nu }\subset\mathbb{R} \times S_{k_1}^{\nu }$.
This shows that the number of zeros of functions $u $ lying in the
projection of $\Gamma _k^{\nu }$ onto the space $E$ is
locally constant, so it is constant and it is equal to $( k-1) $.
Thus, $\Gamma _k^{i}\subset \mathbb{R}\times S_k$.
Set
\[
\Gamma _k^{+}=( \Gamma _k^{1}\cap \mathbb{R}\times S_k^{+})
\cup ( \Gamma _k^{2}\cap \mathbb{R}
\times S_k^{+}) \text{and}\ \Gamma _k^{-}=( \Gamma
_k^{1}\cap \mathbb{R}\times S_k^{-}) \cup ( \Gamma _k^{2}\cap
\mathbb{R}\times S_k^{-})
\]
and let $\varsigma >0$ and $\varkappa \in ( 0,1) $. We have from
Theorem 1.25 in \cite{rab12} that, for $i=1,2$, there exists a sequence
$ ( \mu _n^{i},u_n^{i}) _{n\geq 1}\subset \Gamma _k^{i}$ such
that $| \mu _n^{i}-\mu _k( q_0) | <\varsigma $,
$u_n^{i}=t_n^{i}v_k+w_n^{i},\lim t_n^{i}=0$ and $
t_n^{1}>\varkappa \| u_n^{1}\| ,\ t_n^{2}<-\varkappa
\| u_n^{2}\| $.
 Moreover,  from \cite[Lemma 1.24]{rab12}  we have that
$w_n^{i}=o( | t_n^{i}| ) $.
Arguing as above, we see that
$\lim ( u_n^{i}/\| u_n^{i}\| _{\infty }) =v^{i}$ (up to a subsequence) where
$v^{1}$ and $v^{2}$ are eigenvectors associated with
$\mu _k(q_0) $ with $v^{1}>0$ near $\xi $ and $v^{2}<0$ near $\xi $.
So $v^{1}\in S_k^{+}$ and $v^{2}\in S_k^{-}$. Since the limits are in
$E=C [ \xi ,\eta ] $, arguing as in the proof of existence of
neighborhood $V_k^{i}$, in the beginning of the proof, we obtain that
$u_n^{1}\in S_k^{+}$ and $u_n^{2}\in S_k^{-}$ for $n$ large enough.
This shows that for all $k\geq 3$ and $\nu =+$ or $-$,
$\Gamma _k^{\nu}\neq \emptyset $, and again because of the topology of
$E$ (if $v_n\to v$ in $E$ and $v>0$ near $\xi $ then $v_n>0$ near
$\xi $ for $n$ large) and functions $u$ lying in the projection of
$\Gamma_k^{\nu }$ onto the space $E$ have only simple zeros and all
have the same number of zeros, $\Gamma _k^{\nu }$ does not leave
$\mathbb{R}\times S_k^{\nu }$.

Finally, taking into consideration the claim in the beginning of the proof,
and the fact that $\Gamma _k^{\nu }$ does not leave
$\mathbb{R}\times S_k^{\nu }$ we understand that for all $k\geq 3$ and
$\nu =+$ or $-$, $\Gamma _k^{\nu }$ is unbounded in
$\mathbb{R}\times E$.
\end{proof}

\begin{lemma}\label{4.2}
Assume that \eqref{hyp1}-\eqref{hyp4} hold. Then for all $k\geq 3 $ and
$\nu =+$ or $-$, the component $\Gamma _k^{\nu }$ rejoins the
point $( \chi _k^{\nu },\infty ) $.
\end{lemma}

\begin{proof}
Because $u\equiv 0$ is the unique solution of bvp \eqref{bvp400} for
$\mu =0, $ we have from Lemma \ref{4.1} that for all $k\geq 1$ and
$\nu =+$ or $-, $ $( \{ 0\} \times E) \cap \Gamma _k^{\nu
}=\emptyset $. Therefore, if $( \mu ,u) \in \Gamma _k^{\nu }$
then $\mu >0$.

Moreover, if $( \mu ,u) \in \Gamma _k^{\nu }$, then
$\mu =\mu _k( \frac{f(t,u)}{u},[ \xi ,\eta ] ) $, and this
together with Hypothesis \eqref{hyp4} and Property 3 of Theorem \ref{thm2},
leads to
\[
\mu =\mu _k\Big( \frac{f(t,u)}{u},[ \xi ,\eta ] \Big) \leq
\mu _k( \delta _{\infty },[ \xi ,\eta ] ) .
\]
This shows that for all $k\geq 1$ and $\nu =+$ or $-$, the projection of
$\Gamma _k^{\nu }$ onto the real axis is bounded.

Now, let $(\mu _n,u_n)$ be a sequence in $\Gamma _k^{\nu }$ such that
$\lim_{n\to +\infty }\| u_n\| =+\infty $. For
contradiction purposes, suppose that
$\lim_{n\to +\infty }\mu _n\neq \chi _k^{\nu }$. Then there exist
$\varepsilon >0$ and a subsequence of $(\mu _n)$, which will be denoted
for convenience by $(\mu _n)_{n\geq 1}$, such that
\[
| \mu _n-\chi _k^{\nu }| \geq \varepsilon .
\]

Denote by $(v_n)$ the sequence defined by
$v_n=\frac{u_n}{\| u_n\| }$. Note that $\| v_n\| _{\infty }=1$
and $( \mu _n,v_n) $ satisfies
\[
v_n=\mu _nA_{\infty }v_n+\frac{\Omega u_n}{\|u_n\| _{\infty }}
\]
where $A,K:E\to E$ are defined by
\begin{gather*}
A_{\infty }u( t) =\int_{\xi }^{\eta }G( t,s) (
\alpha _{\infty }( s) u^{+}( s) -\beta _{\infty
}( s) u^{-}( s) ) ds, \\
\Omega u( t) = \int_{\xi }^{\eta }G( t,s) g_{\infty}( s,u( s) ) ds,
\end{gather*}
with $g_{\infty }( t,x) =f( t,x) -\alpha_{\infty }( t) x^{+}
+\beta _{\infty }( t) x^{-}$.
Note that Hypotheses \eqref{hyp3} and \eqref{hyp4} imply that
$\Omega u_n=o( \| u_n\| _{\infty }) $ at $\infty $.
Indeed, we have
\begin{align*}
\frac{| \Omega u_n( t) | }{\|u_n\| _{\infty }}
&= \big| \int_{\xi }^{\eta }G(
t,s) ( \frac{g_{\infty }( s,u_n( s) ) }{
\| u_n\| _{\infty }}) ds\big| \\
&\leq \int_{\xi }^{\eta }G( s,s) \big| \frac{f(
s,u_n( s) ) -\alpha _{\infty }( s)
u_n^{+}( s) +\beta _{\infty }( s) u_n^{-}(
s) }{\| u_n\| _{\infty }}\big| ds.
\end{align*}
From \eqref{hyp3} we have
\begin{align*}
&G( s,s) \big| \frac{f( s,u_n( s)
) -\alpha _{\infty }( s) u_n^{+}( s) +\beta
_{\infty }( s) u_n^{-}( s) }{\|
u_n\| _{\infty }}\big| \\
&\leq G( s,s) ( \gamma _{\infty }( s) +\delta
_{\infty }( s) +\alpha _{\infty }( s) +\beta _{\infty
}( s) ) \frac{| u_n( s) | }{
\| u_n\| _{\infty }} \\
&\leq G( s,s) ( \gamma _{\infty }( s) +\delta
_{\infty }( s) +\alpha _{\infty }( s) +\beta _{\infty
}( s) ) \in L^{1}[ \xi ,\eta ] .
\end{align*}
Now, set
\[
P_n( s) =G( s,s) \big| \frac{f(
s,u_n( s) ) -\alpha _{\infty }( s)
u_n^{+}( s) +\beta _{\infty }( s) u_n^{-}(
s) }{\| u_n\| _{\infty }}\big|
\]
and let us prove that $\lim P_n( s) =0$ for
$s\in [ \xi,\eta ]$  a.e..

Let $s\in [ \xi ,\eta ] $ (such $s$ exists a.e.), such that
\[
\lim_{x\to +\infty }\frac{f( s,x) }{x}=\alpha _{\infty
}( s) \quad \text{and}\quad
\lim_{x\to -\infty }\frac{f(
s,x) }{x}=\beta _{\infty }( s) .
\]
For such an $s$ we distinguish the following cases:

$\bullet$  $\lim u_n( s) =+\infty$: in this case we have
\begin{align*}
P_n( s) &=G(s,s)| \frac{f(s,u_n( s) )}{
u_n( s) }-\alpha _{\infty }( s) | \frac{
u_n( s) }{\| u_n\| _{\infty }} \\
&\leq G(s,s)| \frac{f(s,u_n( s) )}{u_n(
s) }-\alpha _{\infty }( s) | \to 0\quad
\text{as } n\to +\infty .
\end{align*}

$\bullet$ $\lim u_n( s) =-\infty$: in this case we have
\begin{align*}
P_n( s) &=G(s,s)| \frac{f(s,u_n( s) )}{
u_n( s) }-\beta _{\infty }( s) | \frac{
| u_n( s) | }{\| u_n\|
_{\infty }} \\
&\leq G(s,s)| \frac{f(s,u_n( s) )}{u_n(
s) }-\beta _{\infty }( s) | \to 0\quad
\text{as } n\to +\infty .
\end{align*}

$\bullet$ $\lim u_n( s) \neq \pm \infty$:
 in this case there may exist subsequences $( u_{n_k^{1}}( s) ) $ and
$( u_{n_k^{2}}( s) ) $ such that $(u_{n_k^{1}}( s) ) $ is bounded
and $\lim u_{n_k^{2}}( s) =\pm \infty $. Arguing as in the above two
cases we obtain $\lim P_{n_k^{2}}( s) =0$ and we have
\[
P_{n_k^{1}}( s) \leq G( s,s) ( \gamma _{\infty
}( s) +\delta _{\infty }( s) +\alpha _{\infty }(
s) +\beta _{\infty }( s) ) \frac{|
u_{n_k^{1}}( s) | }{\|
u_{n_k^{1}}\| _{\infty }}\to 0\ \text{as}\ k\to
+\infty .
\]

Thus, we have $\lim P_n(s)=0$ for $s\in [ \xi ,\eta ]$  a.e.
By the Lebesgue dominated convergence theorem, we conclude that
$\Omega u_n=o( \| u_n\| _{\infty }) $ at $\infty $.

Now, because of the compactness of $A_{\infty }$ and the boundedness of
$( v_n) $, there exists a subsequence $( v_{n_{j}}) $
converging to $v\in S_k^{\nu }$ (use the same arguments as in the proof of
Lemma \ref{4.1}) with $\| v\| _{\infty }=1$ satisfying
$v=\chi _{\infty }Av$, where $\chi _{\infty }$ is the limit of some
subsequence of $( \mu _{n_{l}}) $ of $( \mu _n) $.
Thus, we have $\chi _{\infty }=\chi _k^{\nu }$ and the contradiction
\[
0=\lim | \mu _{n_{l}}-\chi _k^{\nu }| \geq \varepsilon>0.
\]
\end{proof}

Now, we are able to prove Theorem \ref{thm3}.
Note that $u\in S_i^{\nu }$ is a solution to \eqref{bvp1}
if and only if $( 1,u) \in \Gamma _i^{\nu }$, and this occurs if
$\lambda _i^{\nu }<1<\mu _i(q_0) $ or $\mu _i( q_0) <1<\lambda _i^{\nu }$.

Assume that $\mu _{l}( q_0) <1<\mu _k( \theta _{\infty}) $ with $2<k<l$,
and let $i\in \{ k,\ldots ,l\} $. We have $\mu _i( q_0) \leq \mu _{l}( q_0) <1$,
and from the nondecreasing property of $\lambda _i^{\nu }$ with respect to the
functions $\alpha $ and $\beta $,
\[
\lambda _i^{\nu }=\lambda _i^{\nu }( \alpha _{\infty },\beta
_{\infty }) \geq \lambda _i^{\nu }( \theta _{\infty },\theta
_{\infty }) =\mu _i( \theta _{\infty }) \geq \mu
_k( \theta _{\infty }) >1.
\]
Now, assume that $\mu _{l}( \vartheta _{\infty }) <1<\mu _k( q_0) $
 with $2<k<l$, and let $i\in \{ k,\ldots,l\} $. We have
$\mu _i( q_0) \geq \mu _k(q_0) >1$, and from the nondecreasing property of
$\lambda _i^{\nu} $ with respect to the functions $\alpha $ and $\beta $,
\[
\lambda _i^{\nu }=\lambda _i^{\nu }( \alpha _{\infty },\beta
_{\infty }) \leq \lambda _i^{\nu }( \vartheta _{\infty
},\vartheta _{\infty }) =\mu _i( \vartheta _{\infty })
\geq \mu _k( \vartheta _{\infty }) >1.
\]
Thus, Theorem \ref{thm3} is proved.

\begin{remark} \label{rmk4.20} \rm
Note that if $q_0\in K_{G}\cap L^{1}[ \xi ,\eta ]$,
from Lemma \ref{multalgintegrable} we have that for all $n\geq 1$,
 $\mu _n( q_0) $ is of algebraic multiplicity one.
 Thus, Theorem \ref{thm3}
and Corollary \ref{coro1} can be extended to the case $1\leq k<l$.
\end{remark}

\begin{remark}\label{remfinal} \rm
Theorem \ref{thm3} holds if we replace Hypothesis \eqref{hyp3}
 by the following assumptions:
\begin{gather*}
| f(t,u)-q_0( t) u| \leq \hat{g}_0(t,| u| ) \quad \text{for all }
 | u| \leq \varsigma _0,,\; t\in [ \xi ,\eta ] \text{ a.e., for some }
\varsigma _0>0, \\
| f(t,u)-\alpha _{\infty }(t)u^{+}+\beta _{\infty
}(t)u^{-}| \leq \hat{g}_{\infty }( t,| u|
) \quad \text{for all } u\in\mathbb{R},\; t\in [ \xi ,\eta ]
\text{ a.e.,}\\
\lim_{u\to \varrho }\frac{\hat{g}_{\varrho }(t,u)}{u}=0\quad
\text{in }L_{G}^{1}[ \xi ,\eta ]  \text{ for } \varrho =0,+\infty
\end{gather*}
where $\hat{g}_0,\hat{g}_{+\infty }:[ \xi ,\eta ] \times\mathbb{R}^{+}\to
\mathbb{R}^{+}$ are such that for $\varrho =0,+\infty $,
$\hat{g}_{\varrho },
(\cdot ,u)\in L_{G}^{1}[ \xi ,\eta ]$
 for $u$ fixed and $\hat{g}_{\varrho }( t,\cdot )$
 is  nondecreasing for $t\in [ \xi ,\eta ] $  a.e. Indeed, we have for
$(u_n) \subset E$ with $\lim \| u_n\| _{\infty }=0$
\begin{align*}
\frac{| Hu_n( t) | }{\|u_n\| _{\infty }}
 &\leq \int_{\xi }^{\eta }G( s,s) \frac{| f( s,u_n( s) ) -q_0(
s) u_n( s) | }{\| u_n\|
_{\infty }}ds \\
&\leq \int_{\xi }^{\eta }G( s,s) \frac{ \hat{g}_0 (s,\| u_n\| _{\infty })}
{\|u_n\| _{\infty }}ds\to 0\quad \text{as }n\to +\infty
\end{align*}
leading to $Hu=o ( \| u\| _{\infty }) $ at $0$.

Also,  for $( u_n) \subset E$ with $\lim \| u_n\| _{\infty }=+\infty $
we have
\begin{align*}
\frac{| \Omega u_n( t) | }{\|
u_n\| _{\infty }}
&\leq \int_{\xi }^{\eta }G( s,s)
\frac{| f( s,u_n( s) ) -\alpha _{\infty
}( s) u_n^{+}( s) +\beta _{\infty }( s)
u_n^{-}( s) | }{\| u_n\| _{\infty }}ds \\
&\leq \int_{\xi }^{\eta }G( s,s) \frac{ \hat{g}
_{+\infty }{ (s,\| u_n\| _{\infty })}}{
\| u_n\| _{\infty }}ds\to 0\quad \text{as }n\to +\infty
\end{align*}
leading to $\Omega u=o ( \| u\| _{\infty }) $ at $\infty $.

The function $f$ given in Example \ref{examp1} satisfies the above
condition with
\begin{gather*}
q_0( t) =At^{-3/2}( 1-t) ^{-5/4}, \\
\alpha _{\infty }( t) =At^{-3/2}( 1-t) ^{-5/4}+Bt^{-7/6}( 1-t) ^{-7/4}, \\
\beta _{\infty }( t) =At^{-3/2}( 1-t) ^{-5/4}+Ct^{-11/7}( 1-t) ^{-13/10},
\end{gather*}
and
\begin{gather*}
\hat{g}_0( t,u) =( Bt^{-7/6}( 1-t) ^{-7/4}+Ct^{-11/7}( 1-t) ^{-13/10})u^{3}, \\
\hat{g}_{\infty }( t,u)
=\begin{cases}
MCt^{-11/7}( 1-t) ^{-13/10}+Bt^{-\frac{7}{6}}( 1-t) ^{-7/4},& \text{for } u>0, \\
Ct^{-11/7}( 1-t) ^{-13/10}+MBt^{-7/6}( 1-t) ^{-7/4}, & \text{for }u<0,
\end{cases}
\end{gather*}
where
\[
M=\sup \{ \frac{x^{3}}{1+x^{2}+e^{x}}: x\geq 0\} .
\]
\end{remark}



\subsection*{Acknowledgements}
The authors want to thank the anonymous referee for the
comments and suggestions about this article.

 A. Benmeza\"{\i} and W. Esserhane, would like
to thank their laboratory, Fixed Point Theory and Applications,
for supporting this work.

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\end{document}