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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 245, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/245\hfil Positive solutions]
{Positive solutions for second-order boundary-value problems with sign
changing Green's functions}

\author[A. Cabada, R. Engui\c{c}a, L. L\'opez-Somoza \hfil EJDE-2017/245\hfilneg]
{Alberto Cabada, Ricardo Engui\c{c}a, Luc\'ia L\'opez-Somoza}

\address{Alberto Cabada \newline
Instituto de Matem\'aticas,
Facultade de Matem\'aticas,
Universidade de Santiago de Compostela, 15782,
Santiago de Compostela, Galicia, Spain}
\email{alberto.cabada@usc.es}

\address{Ricardo Engui\c{c}a \newline
Departamento de Matem\'atica,
Instituto Polit\'ecnico de Lisboa,
 Lisboa, Portugal}
\email{rroque@adm.isel.pt}

\address{Luc\'{\i}a L\'opez-Somoza \newline
Instituto de Matem\'aticas,
Facultade de Matem\'aticas,
Universidade de Santiago de Compostela, 15782,
Santiago de Compostela, Galicia, Spain}
\email{lucia.lopez.somoza@usc.es}

\dedicatory{Communicated by  Pavel Drabek}

\thanks{Submitted June 19, 2017. Published October 6, 2017.}
\subjclass[2010]{34B15, 34A40}
\keywords{Second order differential equations; Dirichlet boundary conditions;
\hfill\break\indent  periodic boundary conditions; sign changing Green's function}

\begin{abstract}
 In this article we analyze some possibilities of finding positive
 solutions for second-order boundary-value problems with the Dirichlet
 and periodic  boundary conditions, for which the corresponding Green's
 functions change sign. The obtained results can also be adapted to
 Neumann and mixed boundary conditions.
\end{abstract}

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

 \section{Introduction}

 In the literature, the existence of positive solutions for boundary-value
problems (BVP) has been widely studied, in particular for second-order BVP with
periodic and Dirichlet boundary conditions.
A standard technique consists in obtaining the existence of positive
solutions through Krasnoselskii's fixed point theorem on cones,
or to use fixed point index theory. In these cases, the positivity
of the associated Green's functions is usually fundamental to prove
such results. In this paper we are able to prove existence of solutions
for several problems where the associated Green's function changes sign.

 Hill's operator properties have been described in several papers, where
existence and multiplicity results, comparison principles, Green's functions
and spectral analysis were studied. Some of these results can be originally
found in \cite{cacidAAA, cacid, cacilu, torres, zhang}.

 Positivity results for BVP where the Green's function can vanish are
treated for example in \cite{gkw1, webb}.  Graef, Kong and Wang \cite{gkw1}
studied the periodic BVP (with $T=1$)
\begin{gather*}
 u''(t)+a(t)\,u(t)=g(t)f(u(t)), \quad t\in(0,T),\\
 u(0)=u(T), \quad u'(0)=u'(T),
\end{gather*}
 with $f$ and $g$ nonnegative continuous functions and $g$ satisfying the
condition $\min_{t\in[0,T]}g(t)>0$. They assumed the Green's function to be
nonnegative and to satisfy the  condition
 \begin{equation}\label{int-graef}
 \min_{0\le s\le T} \int_0^T G(t,s) \, dt>0.
 \end{equation}
Webb \cite{webb} considered weaker assumptions to prove the existence of
positive solutions of the previous problem, but he still assumed  Green's
function to be nonnegative. Despite our results do not require the Green's
 function to be nonnegative, they could be applied to this particular case,
obtaining positive solutions assuming an integral condition weaker
than \eqref{int-graef} (see Remarks \ref{R-int-Graef} and \ref{R-int-Graef2}).

 On the other hand, some existence results for BVP with sign-changing Green's
function have been considered in \cite{cabinftoj,infpietoj},
 where the authors asked for the existence of a subinterval
 $[c,d]\subset [0,T]$, a function $\phi\in L^1([0,T])$ and a constant
$c\in(0,1]$ such that the Green's function $G$ satisfies the  condition
 \begin{equation}\label{G-banda}
\begin{gathered}
 |G(t,s)|\le \phi(s) \text{ for all } t\in[0,T] \text{ and almost every }
s\in[0,T],\\
 G(t,s)\ge c\, \phi(s) \text{ for all } t\in[c,d] \text{ and almost every }
 s\in[0,T].
 \end{gathered}
\end{equation}

It must be pointed out that, if we consider a periodic problem with constant
potential $a(t)=\rho^2$ for which the related Green's function changes its
 sign (i.e. $\rho > \pi/T$, $\rho \neq 2 k \pi/T$, $k=1,2, \ldots$),
condition \eqref{G-banda} is never fulfilled for any strictly positive
function $\phi$. This is due to the fact that in such situation the Green's
function is constant along the straight lines of slope equals to one
(see \cite{cabada1, cabada2} for details). Meanwhile, as we will prove on
Section \ref{sect_ex_per}, our results can be applied without further
complications for this case.

 Moreover, for the Dirichlet BVP with constant potential $a(t)=\rho^2$
with sign-changing Green's function (i.e. $\rho > \pi/T$, $\rho \neq k \pi/T$,
$k=1,2, \ldots$), as a direct consequence of expression \eqref{e-G-Dir} below,
it is immediate to verify that condition \eqref{G-banda} holds if and only
if $\rho^2$ lies between the first and the second eigenvalues of the problem
($\frac{\pi}{T}<\rho<\frac{2 \pi}{T}$) but it is never satisfied for
 $\rho>\frac{2 \pi}{T}$. However, as we will point out in
Section \ref{sect-Dir-constant}, our results can be applied for any
 nonresonant value of $\rho > \pi/T$. Despite this, we must note that
the imposed restrictions increase with $\rho$.

 Furthermore, in \cite{cabinftoj,infpietoj} the authors proved the existence
of solutions in the cone
 \[
K_0=\big\{u\in\mathcal{C}[0,T]: \min_{t\in[c,d]} u(t)\ge c \, \|u\|\big\},
\]
 that is, they ensured the positivity of the solutions on the subinterval
$[c,d]$ but such solutions were allowed to change sign when considering
the whole interval $[0,T]$.

 As far as we know, positive solutions for BVP with sign-changing Green's
function can be tracked only as back as 2011 in the papers \cite{ma, ZA}.
In the first of these papers,  Ma considers the  one-parameter
family of problems
 \begin{equation} \label{e-per-lambda}
\begin{gathered}
 u''(t)+a(t)\,u(t)=\lambda\,g(t) f(u(t)), \quad t\in(0,T),\\
 u(0)=u(T), \quad u'(0)=u'(T).
\end{gathered}
\end{equation}
By using the Schauder's fixed point Theorem, the author obtains the existence
of a positive solution for sufficiently small values of $\lambda$.
These existence results are not comparable with the ones we will obtain
in this paper.
In the second paper,  Zhong and  An \cite{ZA} study the following autonomous periodic BVP,
 with constant potential $\rho \in (0, \frac{3  \pi}{2\, T}]$:
 \begin{equation}
 \label{e-per-Zhong}
 u''+\rho^2 u=f(u),\; t\in(0,T),\quad u(0)=u(T), \; u'(0)=u'(T).
 \end{equation}
In this case, it is very well known that the related Green's function
$G_P(t,s) \ge 0$ for all $\rho \in (0, \frac{\pi}{T}]$ and it changes
sign for $\rho \in (\frac{\pi}{T}, \frac{3  \pi}{2\, T}]$
(see \cite{cabada1, cabada2}).
With this, it can be defined the constant
 \begin{equation*}
 \delta=\begin{cases}
 \infty &\text{if }  \rho \in (0, \frac{\pi}{T}],\\[4pt]
 \inf_{t\in I} \frac{\int_{0}^{T} G^+_P(t,s)\,ds}
{\int_{0}^{T} G^-_P(t,s)\,ds} &\text{if}
 \rho \in (\frac{\pi}{T}, \frac{3  \pi}{2 T}],
 \end{cases}
\end{equation*}
and using the Krasnoselskii's fixed point Theorem, the authors prove
the following existence result:

\begin{theorem}\cite[Theorem 3]{ZA} \label{t-ZA}
Suppose that the following assumptions are fulfilled:
\begin{enumerate}
\item  $f:[0, \infty) \to [0, \infty) $ is continuous.
\item  $0 \le m = \inf_{u \ge 0}{\{f(u)\}}$ and
 $M= \sup_{u \ge 0}{\{f(u)\}} \le M \le \infty$.
\item  $M/m \le \delta$, with $M/m = \infty$ when $m=0$.
\end{enumerate}
Moreover, if $\delta=\infty$ assume that
\[
 \lim_{x\to \infty}  \frac{f(x)}{x}< \rho^2 < \lim_{x\to 0^+} \frac{f(x)}{x}.
\]
Then problem \eqref{e-per-Zhong} has a positive solution on $[0,T]$.
\end{theorem}

 Concerning this specific case, along this paper we improve the range
of the values $\rho$ for which the result is still valid.
Furthermore, we apply our study to nonconstant potentials and nonautonomous
nonlinear parts.

As we will see, some of the positivity conditions imposed for the periodic
BVP cannot be adapted for the Dirichlet BVP, so the approach that must
be used needs to be considerably modified, by using, in this case,
a different type of cones.

The rest of this article is organized the following way:
In Section \ref{sect-preliminaries} we state some preliminary results
considering the Hill's operator.
In Section \ref{sect-periodic} some new results concerning the existence
of a positive solution for the Hill's periodic BVP in the case that
 the Green's function may change sign are proved.
 Moreover, in this section, such existence results are generalized
to other boundary conditions.
In Section \ref{sect_ex_per} we improve Theorem \ref{t-ZA} for the periodic
problem with a constant potential.
In Section \ref{sect-Dir-constant} we approach the Dirichlet BVP,
also in the case of a constant potential, where as far as we know,
no results for sign changing Green's function were proved before.

 \section{Preliminaries}  \label{sect-preliminaries}

 Let $L[a]$ be the Hill's operator related to the potential $a$
 \begin{equation*}
 L[a]\,u(t)\equiv u''(t)+ a(t)\,u(t), \quad t\in [0,T]\equiv I,
 \end{equation*}
where $a\colon I\to \mathbb{R}$, $a\in L^\alpha( I)$, $\alpha\ge 1$.

 Let $X \subset W^{2,1}( I)$ be a Banach space such that the homogeneous problem
 \begin{equation}  \label{e-Hill-X}
 L[a]\,u(t)=0, \quad  \text{for a.\,e. }  t\in I, \quad u\in X
 \end{equation}
has only the trivial solution. This condition is known as operator
$L[a]$ being nonresonant in $X$.
Moreover, it is very well known that if this condition is satisfied and
$\sigma \in L^1( I)$, the nonhomogeneous problem
 \begin{equation*}
 L[a]u(t)=\sigma(t), \quad \text{for a.\,e. }  t\in I, \quad u\in X
 \end{equation*}
has a unique solution
 \begin{equation*}
 u(t)=\int_{0}^{T} G(t,s)\sigma(s)\, ds, \quad t\in I,
 \end{equation*}
where $G$ is the corresponding Green's function.

 We denote $x \succ 0$ on $I$ if  $x \ge 0$ on $I$ and $\int_0^T{x(s) \, ds } >0$.
It is said that operator $L[a]$ satisfies a strong maximum principle (MP)
in $X$ if
 \begin{equation*}
 u\in X, \; L[a]\,u \succ 0 \text{ on } I\;\Rightarrow\;
 u < 0 \text{ in } (0,T).
 \end{equation*}
Analogously, $L[a]$ satisfies the antimaximum principle (AMP) in $X$ if
\begin{equation*}
 u\in X, \; L[a]\,u \succ 0 \text{ on } I\;\Rightarrow\; u > 0 \text{ in } (0,T).
\end{equation*}

The next result is a direct consequence of
\cite[Corollaries 1.6.6 and 1.6.12]{cabada2}, and it ensures that the
maximum and anti-maximum principles for the periodic problem are equivalent
to the constant sign of the Green's function.

 \begin{lemma}  \label{l-Green-comparison}
 The following claims are equivalent:
 \begin{itemize}

 \item[(1)] The related Green's function $G$ of problem \eqref{e-Hill-X}
satisfies $G (t,s) \ge 0$ $(\le 0)$ on $ I\times I$.

 \item[(2)] Operator $L[a]$ satisfies a strong maximum (antimaximum)
principle in $X$.
\end{itemize}
 \end{lemma}


 We will consider now the periodic boundary-value problem
 \begin{equation}\label{e-P} %\tag{$P$}
\begin{split}
 u''(t)+a(t)\,u(t)=0, \; t\in I, \quad u(0)=u(T),\; u'(0)=u'(T),
 \end{split}\end{equation}
 and we will denote its related Green's function as $G_P$.

 Now, let ${\lambda}_P$ be the smallest eigenvalue of the periodic problem
 $$
 u''(t)+(a(t) + \lambda)\, u(t)=0,\quad \text{ for a.\,e. }\;t\in I, \quad
 u(0)=u(T),\; u'(0)=u'(T),
 $$
 and let ${\lambda}_A$ be the smallest eigenvalue of the anti-periodic problem
 $$
 u''(t)+(a(t) + \lambda)\, u(t)=0,\quad \text{ for a.\,e. }\;t\in I,
 \quad u(0)=-u(T),\; u'(0)=-u'(T).
 $$
 In \cite{zhang} it is proved that ${\lambda}_P<{\lambda}_A$.
 The following result relates the constant sign of the periodic Green's
function with the sign of these eigenvalues:

 \begin{lemma} \cite[Theorem 1.1]{zhang} \label{l-zhang-eigen}
 Suppose that $a \in L^1( I)$, then:
 \begin{enumerate}
 \item $G_P(t,s) \le 0$ on $ I\times I$ if and only if ${\lambda}_P>0$.
 \item $G_P(t,s) \ge 0$ on $ I\times I$ if and only if
${\lambda}_P< 0\le {\lambda}_A$.
 \end{enumerate}
 \end{lemma}

If we consider other boundary-value problems, such as the Neumann problem
 \begin{equation}\label{Neumann} %\tag{$N$}
 u''(t)+a(t)\,u(t)=0, \; t\in I, \quad u'(0)=u'(T)=0;
 \end{equation}
 the Dirichlet problem
 \begin{equation}\label{Dirichlet} %\tag{$D$}
 u''(t)+a(t)\,u(t)=0, \; t\in I,\quad u(0)=u(T)=0;
 \end{equation}
 and the mixed problems
 \begin{gather}\label{Mixed1} %\tag{$M_1$}
 u''(t)+a(t)\,u(t)=0, \; t\in I,\quad u'(0)=u(T)=0;\\
\label{Mixed2}  %\tag{$M_2$}
 u''(t)+a(t)\,u(t)=0, \; t\in I,\quad u(0)=u'(T)=0;
\end{gather}
 denoting by $G_N$, $G_D$, $G_{M_1}$ and $G_{M_2}$ the related Green's
functions and $\lambda_N$, $\lambda_D$, $\lambda_{M_1}$ and
$\lambda_{M_2}$ the corresponding smallest eigenvalue of each of the problems,
we know that the following results are satisfied (see \cite{cacilu}):

 \begin{lemma}  \label{l-Meu-Dir-Mix}
 \begin{enumerate}
 \item $G_N(t,s)<0$ on $ I\times I$ if and only if $\lambda_N>0$.

 \item $G_N(t,s)\ge 0$ on $ I\times I$ if and only if
$\lambda_N<0$, $\lambda_{M_1}\ge 0$ and $\lambda_{M_2}\ge 0$.

 \item $G_N$ changes sign if and only if $\min\{\lambda_{M_1},\,\lambda_{M_2}\}< 0$.

 \item $G_D(t,s)<0$ on $(0,T)\times(0,T)$ if and only if $\lambda_D>0$.

 \item $G_D$ changes sign if and only if $\lambda_D<0$.

 \item $G_{M_1}(t,s)<0$ on $[0,T)\times[0,T)$ if and only if $\lambda_{M_1}>0$.

 \item $G_{M_1}$ changes sign if and only if $\lambda_{M_1}<0$.

 \item $G_{M_2}(t,s)<0$ on $(0,T]\times(0,T]$ if and only if $\lambda_{M_2}>0$.

 \item $G_{M_2}$ changes sign if and only if $\lambda_{M_2}<0$.
 \end{enumerate}
 \end{lemma}


 \section{Periodic boundary-value problems}  \label{sect-periodic}

 Consider now the  nonlinear and nonautonomous periodic boundary
value problem
 \begin{equation}\label{hill_periodic}\begin{split}
 u''(t)+a(t)\,u(t)=f(t,u(t)), \; t\in I, \quad u(0)=u(T), \; u'(0)=u'(T).
 \end{split}\end{equation}
We will assume that problem \eqref{e-P} is nonresonant and ${\lambda}_A< 0$.
From Lemma \ref{l-zhang-eigen}, it is clear that in this case the related
Green's function changes its sign on $I \times I$.

 On the other hand, it is well-known that there exists $v_P$, a positive
eigenfunction on $I$, unique up to a constant, related to $\lambda_P$;
 that is, $v_P$ is such that
 \begin{gather*}
  v_P''(t)+a(t) v_P(t)=-\lambda_P v_P(t), \quad \text{ a.\,e. }  t\in I,\\
  v_P(0)=v_P(T), \quad  v_P'(0)=v_P'(T).
\end{gather*}
 Therefore,
 \begin{equation*}
 v_P(t)=-\lambda_P\int_{0}^{T} G_P(t,s) v_P(s)\,ds
 \end{equation*}
and, since $v_P$ is positive and $\lambda_P<0$, we have that
 \begin{equation*}
 \int_{0}^{T} G_P(t,s)v_P(s)\,ds>0 \quad \forall t\in I
 \end{equation*}
and, consequently,
 \begin{equation*}
 \int_{0}^{T} G^+_P(t,s)v_P(s)\,ds>\int_{0}^{T} G^-_P(t,s)v_P(s)\,ds \quad
\forall t\in I,
 \end{equation*}
where $G^+_P$ and $G^-_P$ are the positive and negative parts of $G_P$.

 Since the Green's function changes sign, it makes sense to define
 \begin{equation*}
 \gamma= \inf_{t\in I} \frac{\int_{0}^{T} G^+_P(t,s) v_P(s)\,ds}
{\int_{0}^{T} G^-_P(t,s) v_P(s)\,ds}\quad (>1) .
 \end{equation*}
Moreover,  to ensure the existence of solutions of problem \eqref{hill_periodic},
we will make the following assumptions:
 \begin{enumerate}
 \item[(H1)] $f\colon I\times [0,\infty) \to [0,\infty)$ satisfies
$L^1$-Carath\'eodory conditions, that is, $f(\cdot,u)$ is measurable for every
$u\in\mathbb{R}$, $f(t,\cdot)$ is continuous for a.\,e. $t\in I$ and for each
$r>0$ there exists $\phi_r \in L^1 (I)$ such that $f(t,u) \le \phi_r(t)$
 for all $u\in[-r,r]$ and a.\,e. $t\in I$.

 \item[(H2)] There exist two positive constants $m$ and $M$ such that
$m v_P(t)\le f(t,x) \le M v_P(t)$ for every $t\in I$ and $x\ge 0$.
Moreover, these constants satisfy that $\frac{M}{m}\le \gamma$.

 \item[(H3)] There exists $[c,d]\subset I$ such that $\int_{c}^{d} G_P(t,s) \,dt
\ge 0$, for all $s\in I$ and $\int_{c}^{d} G_P(t,s) \,dt > 0$,
for all $s\in [c,d]$.
\end{enumerate}

 \begin{remark}  \label{r-condi-per} \rm
 We note that condition (H2) includes, as particular cases, hypotheses
(2) and (3) in Theorem \ref{t-ZA} imposed in \cite{ZA}.
This is so because if $a(t)=\rho^2$, as in problem \eqref{e-per-Zhong},
we have that $\lambda_P=-\rho^2$ and $v_P(t)=1$ for all $t \in I$.
 Moreover, as we will point out in Section \ref{sect_ex_per}, we have that
if $a(t)=\rho^2$ then
 \[
\int_{0}^{T} {G_P(t,s)\,ds}=\frac{1}{\rho^2},
\]
and condition (H3) is trivially fulfilled for $[c,d]=I$.

 Moreover, we note that in (H2) we are not considering the possibility of $m=0$.
Theorem \ref{t-ZA} includes this case, but only when $\gamma=+\infty$, which
only happens when the Green's function is nonnegative. In \cite{ZA}
the authors consider this possibility because they are assuming that
$\rho\in \big( 0, \frac{3 \pi}{2 T}\big]$ and when
 $\rho\in \big( 0, \frac{\pi}{T}\big]$, $G_P$ is nonnegative.
As we will see in Corollary \ref{cor-GP-posit}, hypothesis (H2)
is not necessary in this case, so this is the reason why we do not
consider the possibility $m=0$.
 \end{remark}

 We will consider the Banach space $(\mathcal{C}( I,\mathbb{R}), \, \|\cdot\|)$
coupled with the supremum norm $\|u\|\equiv \|u\|_\infty$, and define the cone
 \[
K=\big\{u\in \mathcal{C}( I,\mathbb{R}): u\ge 0 \text{ on $I$}, \;
\int_{0}^{T}u(s)\,ds \ge \sigma  \|u\| \big\},
\]
 where
 \begin{equation*}
 \sigma = \frac{\eta}{ \max_{t,\,s\in I} \{ G_P(t,s)\}},
 \end{equation*}
with
 \begin{equation}  \label{e-eta}
 \eta =\min_{s\in [c,d]} \big\{\int_{c}^{d}G_P(t,s)\,dt\big\}.
 \end{equation}
Now, it is clear that $u$ is a solution of the periodic problem
\eqref{hill_periodic} if and only if it is a fixed point of the following operator:
 \begin{equation*}
 {\mathcal{T}} u(t)=\int_{0}^{T} G_P(t,s) f(s,u(s)) \,ds.
 \end{equation*}

 \begin{lemma}
 Assume hypothesis {\rm (H1)--(H3)}. Then
${\mathcal{T}} \colon \mathcal{C}( I) \to \mathcal{C}( I)$ is a completely continuous operator
 which maps the cone $K$ to itself.
 \end{lemma}

 \begin{proof}
 The proof that operator ${\mathcal{T}}$ is a completely continuous operator
follows standard arguments and we omit it.

Let us see now that ${\mathcal{T}}$ maps the cone to itself.
Considering $u \in K$, then, for all $t \in I$, the following inequalities
are fulfilled:
 \begin{align*}
 {\mathcal{T}} u(t)
& =\int_{0}^{T}G_P(t,s) f(s,u(s))\,ds \\
&= \int_{0}^{T}\left(G^+_P(t,s)-G^-_P(t,s)\right) f(s,u(s))\,ds \\
 &\ge \int_{0}^{T}\left(m\, v_P(s)\,G^+_P(t,s)-M v_P(s)\,G^-_P(t,s)\right)\,ds \\
 &\ge m \Big( \int_{0}^{T}G^+_P(t,s) v_P(s)\,ds
-\gamma\int_{0}^{T}G^-_P(t,s) v_P(s)\,ds \Big)\ge 0.
\end{align*}
 Moreover,
 \begin{align*}
 \int_{0}^{T} {\mathcal{T}} u(t) \,dt
& \ge \int_{c}^{d} {\mathcal{T}} u(t) \,dt
 = \int_{c}^{d} \int_{0}^{T}G_P(t,s) f(s,u(s))\,ds \,dt\\
& = \int_{0}^{T} f(s,u(s)) \int_{c}^{d} G_P(t,s)\,dt \,ds \\
& \ge \eta  \int_{0}^{T} f(s,u(s))\,ds,
\end{align*}
and since
 \begin{equation*}
 {\mathcal{T}} u(t) \le \max_{t,s\in I} \{G_P(t,s) \}
\int_{0}^{T}f(s,u(s))\,ds,
 \end{equation*}
we deduce that $\int_{0}^{T} {\mathcal{T}} u(t) \,dt \ge \sigma {\mathcal{T}} u(t)$
 for all $t\in I$, that is
 \begin{equation*}
 \int_{0}^{T} {\mathcal{T}} u(t) \,dt \ge \sigma \|{\mathcal{T}} u\|,
 \end{equation*}
and the result is concluded.
 \end{proof}


Now,  to prove the existence of solutions for problem \eqref{hill_periodic},
 we  use some classical results regarding the fixed point index.
 We compile them in the following lemma.
Let $\Omega$ be an open bounded subset of $C(I)$ and let us denote
$\bar \Omega$ and $\partial\Omega$ its closure and boundary, respectively.
 Moreover, let us denote $\Omega_K=\Omega \cap K$.

 \begin{lemma} \cite[Lemma 12.1]{amann2}
 Let $\Omega_K$ be an open bounded set with $0\in\Omega_K$ and
$\bar \Omega_K\neq K$. Assume that $F\colon \bar \Omega_K \to K$ is a
completely continuous map such that $x\neq Fx$ for all $x\in\partial \Omega_K$.
Then the fixed point index $i_K(F,\Omega_K)$ has the following properties:
 \begin{enumerate}
 \item If there exists $e\in K\setminus\{0\}$ such that $x\neq Fx+\lambda e$
for all $x\in\partial \Omega_K$ and all $\lambda>0$, then $i_K(F,\Omega_K)=0$.
 \item If $x\neq \mu\, Fx$ for all $x\in\partial \Omega_K$ and for every
$\mu\le 1$, then $i_K(F,\Omega_K)=1$.
 \item If $i_K(F,\Omega_K)\neq 0$, then $F$ has a fixed point in $\Omega_K$.
 \item Let $\Omega^1_K$ be an open set with $\bar \Omega^1_K\subset \Omega_K$.
If $i_K(F,\Omega_K)=1$ and $i_K(F,\Omega^1_K)=0$, then $F$ has a fixed point
in $\Omega_K \setminus \bar \Omega^1_K$. The same result holds if
$i_K(F,\Omega_K)=0$ and $i_K(F,\Omega^1_K)=1$.
 \end{enumerate}
 \end{lemma}

 Now we are in a position to prove the existence results concerning the periodic
problem \eqref{hill_periodic} as follows. First, we note that, as an
immediate consequence of condition (H2), we deduce the following properties:
 \begin{align*}
 f_0&=\lim_{x\to 0^+} \big\{\min_{t\in [c,d]} \frac{f(t,x)}{x}\big\}
=\infty, \quad f^{\infty}=\lim_{x\to \infty}
\big\{\max_{t\in I} \frac{f(t,x)}{x}\big\}=0,
\end{align*}
 where the interval $[c,d]$ is given in (H3).
These properties will let us prove the following theorem.

 \begin{theorem} \label{thm3.4}
 Assume that $\lambda_A<0$ and hypothesis {\rm (H1)--(H3)} hold.
Then there exists at least one positive solution of problem \eqref{hill_periodic}
in the cone $K$.
 \end{theorem}

 \begin{proof}
Taking into account the definition of $f_0$, we know that there exists
$\delta_1>0$ such that when $\|u\|\le\delta_1$, then
 $$
f(t,u(t))> \frac{u(t)}{\eta}\,, \quad \forall  t\in [c,d],
$$
 with $\eta$ defined in \eqref{e-eta}.
Let
 $$
\Omega_{1}=\{u\in K: \|u\|<\delta_1\}
$$
 and choose $u\in\partial \Omega_1$ and $e\in K\setminus\{0\}$.

We will prove that $u \neq {\mathcal{T}} u+\lambda\,e$ for every $\lambda>0$.
Assume, on the contrary, that there exists some $\lambda>0$ such that
$u={\mathcal{T}} u+\lambda\,e$, that is,
 \begin{equation*}
 u(t)= {\mathcal{T}} u(t)+\lambda\,e(t)\ge {\mathcal{T}} u(t) \quad
\forall\,t\in I.
 \end{equation*}
Then
\begin{align*}
 \int_{c}^{d}u(t)\, dt
&\ge \int_{c}^{d} {\mathcal{T}} u(t)\,dt
 =\int_{c}^{d} \int_{0}^{T} G_P(t,s) f(s,u(s))\,ds\,dt\\
& = \int_{0}^{T} \Big(\int_{c}^{d} G_P(t,s) \,dt\Big) f(s,u(s))\,ds \\
 & \ge \int_{c}^{d} \Big(\int_{c}^{d} G_P(t,s) \,dt\Big) f(s,u(s))\,ds
> \int_{c}^{d}u(s) \,ds,
\end{align*}
 which is a contradiction.
Therefore $i_K(T,\Omega_1)=0$.

 Proceeding in an analogous way to \cite{cacid, gkw1, gkw2}, we define
 $ \tilde{f}(t,u)=\max_{0\le z\le u}{f(t,z)}$. Clearly
$\tilde{f}(t, \cdot)$ is a nondecreasing function on $[0,\infty)$.
Moreover, since $f^{\infty}=0$ it is obvious that
$$
\lim_{x\to \infty}  \big\{\max_{t\in I} \frac{\tilde{f}(t,x)}{x}\big\}=0.
$$
As a consequence,  there exists $\delta_2>0$ such that if $\|u\|\ge\delta_2$ then
 $$
\tilde{f}(t,\|u\|)< \frac{\sigma^2}{T^2 \eta}\|u\| \quad \forall t\in I.
$$
Let
 $$
\Omega_{2}=\{u\in K; \ \|u\|<\delta_2\}
$$
 and choose $u\in \partial\, \Omega_{2}$.

 We will prove that $u \neq \mu  {\mathcal{T}} u$ for every $\mu\le 1$.
Assume, on the contrary, that there exists some $\mu\le 1$ such that
$u(t)=\mu \, {\mathcal{T}} u(t)$ for all $t\in I$. Then
 \begin{align*}
\sigma \|u \|
&\le \int_{0}^{T}u(t)\,dt
 = \mu \int_{0}^{T} {\mathcal{T}} u(t)\,dt \\
&= \mu \int_{0}^{T} \int_{0}^{T} G_P(t,s) f(s,u(s))\,ds\,dt\\
&=\mu \int_{0}^{T} \Big(\int_{0}^{T} G_P(t,s) \,dt\Big)f(s,u(s))\,ds \\
&\le \mu T \max_{t,s\in I} \{G_P(t,s) \} \int_{0}^{T} f(s,u(s))\,ds \\
&\le \mu T \max_{t,s\in I} \{G_P(t,s) \} \int_{0}^{T} \tilde{f}(s,u(s))\,ds \\
&\le \mu T \max_{t,s\in I} \left\{G_P(t,s) \right\} \int_{0}^{T} \tilde{f}(s,\|u\|)\,ds \\
&< \mu T^2 \frac{\eta}{\sigma} \frac{\sigma^2}{T^2 \eta}  \|u\|
 \le \sigma\, \|u \|,
\end{align*}
 which is a contradiction. As a consequence, $i_K(T,\Omega_2)=1$.
We conclude that operator ${\mathcal{T}}$ has a fixed point, that is,
there exists at least a nontrivial solution of problem \eqref{hill_periodic}.
 \end{proof}

The previous theorem is also valid if the Green's function is nonnegative.
 In this case, hypothesis (H3) would be trivially fulfilled and hypothesis
(H2) is not necessary since it is only used to prove that $\mathcal{T}$
maps the cone to itself, which is obvious (since $f$ is nonnegative)
when $G_P$ is nonnegative. On the other hand, we would need to add the
hypothesis that $f_0=\infty$ and $f^\infty=0$ (which can not be deduced if
 we eliminate (H2)).
The result reads as follows:

 \begin{corollary}\label{cor-GP-posit}
 Assume that $\lambda_P < 0 \le \lambda_A$ and hypothesis {\rm (H1)}
 is fulfilled. Then, if $f_0=\infty$ and $f^\infty=0$ there exists at least
one positive solution of problem \eqref{hill_periodic} in the cone $K$.
 \end{corollary}

 \begin{remark}\label{R-int-Graef} \rm
 We note that for a nonnegative Green's function, we generalize the results of Graef, Kong and Wang \cite{gkw1, gkw2} and Webb \cite{webb} since our condition (H3) is weaker than condition \eqref{int-graef} considered by them.
 \end{remark}

\begin{corollary}
If $f(t,x)\equiv f(t)\in L^1(I)$ satisfies (H2), then the unique solution
of \eqref{hill_periodic} is a nonnegative function on $[0,T]$.
\end{corollary}

 \begin{remark}\label{rem_a}\rm
 We note that $u(t)\equiv 1$ is the unique solution of the periodic problem
\begin{gather*}
  u''(t)+a(t)\,u(t)=a(t), \quad t\in I, \\
  u(0)=u(T), \quad u'(0)=u'(T).
\end{gather*}
 Therefore it is clear that
\begin{equation}\label{int-a}
 \int_{0}^{T} G_P(t,s)\,a(s)\,ds=1>0
\end{equation}
 and so the previous reasoning is also valid if $a\ge 0$, $a>0$ on
$[c,d]$, and we change the definition of $\gamma$ by
 \[
\gamma^*= \inf_{t\in I} \frac{\int_{0}^{T} G^+_P(t,s)\,a(s)\,ds}
{\int_{0}^{T} G^-_P(t,s)\,a(s)\,ds}.
\]
 In this case, assumption (H2) would be substituted by
 \begin{itemize}
 \item[(H2')] There exist two positive constants $m$ and $M$ such that
$m a(t)\le f(t,u) \le M a(t)$ for every $t\in I$, $u>0$.
Moreover, these constants satisfy that $\frac{M}{m}\le \gamma^*$.
 \end{itemize}
 \end{remark}

 \subsection{Neumann, Dirichlet and mixed boundary value problems}

 From the classical spectral theory \cite{zettl}, it is very well know that,
as in the periodic case, for any of the boundary conditions introduced in
Lemma~\ref{l-Meu-Dir-Mix}, there exists a positive eigenfunction on $(0,T)$
 related to the corresponding smallest eigenvalue. Therefore,
if we are in the case in which $L[a]$ operator coupled with the associated
boundary conditions is nonresonant and the related Green's function changes
sign (different cases are characterized in Lemma \ref{l-Meu-Dir-Mix}),
we could follow the same argument as in the previous section to define
$\gamma$ and we would obtain analogous existence results. Hypothesis
{\rm (H1)--(H3)} would be the same with the suitable notation for each of
the problems (that is, considering in each case the appropriate Green's
function and eigenfunction).

 \begin{remark} \rm
 For the Neumann problem, it is not difficult to verify that we also have
that if $a(t)=\rho^2$ then
 \[
\int_{0}^{T} {G_N(t,s)\,ds}=\frac{1}{\rho^2},
\]
and condition (H3) is trivially fulfilled for $[c,d]=I$.

 On the other hand, since $u(t)\equiv 1$ is the unique solution of
\begin{align*}
 u''(t)+a(t)\,u(t)=a(t), \; t\in I, \quad
 u'(0)=u'(T)=0,
\end{align*}
 Remark \ref{rem_a} is also valid for the Neumann problem.
\end{remark}

 \begin{remark} \rm
 For the Dirichlet problem, condition (H3) does not hold for $[c,d]=I$.
This is so because  $G_D(t,\cdot)$ satisfies the Dirichlet boundary value
conditions for all $t\in[0,T]$, that is, $G_D(t,0)=G_D(t,T)=0$.

It is important to note that the eigenfunction $v_D$ is positive on
$(0,T)$ but $v_D(0)=v_D(T)=0$, so condition (H2) would imply that
$f(0,x)=f(T,x)=0$ for every $x\ge 0$. However, since as we have mentioned,
$[c,d]\neq I$, this property does not affect on the fact that $f_0=\infty$.

 An analogous situation occurs for the mixed problems. In these cases
it is also impossible to consider $[c,d]=I$ since the corresponding
Green's functions and eigenfunctions vanish on one side of the interval.

 Moreover, if we consider the Dirichlet and mixed problems, the constant
function $u(t)\equiv 1$ is not a solution of the related linear problem
$L[a] u(t)=a(t)$. So, Remark \ref{rem_a} is not longer valid for such
situations.
 \end{remark}

 \begin{remark}\label{R-int-Graef2} \rm
 As it was commented in Remark \ref{R-int-Graef}, we also generalize
the results of Graef, Kong and Wang \cite{gkw1, gkw2} and Webb \cite{webb}
for a nonnegative Green's function coupled with the Neumann conditions.

 Moreover, the results in \cite{gkw1, gkw2, webb} could not be applied to
any Dirichlet problem since the related Green's function will cancel on the
whole lines $s=0$ and $s=T$ so the minimum in \eqref{int-graef} would be 0,
however our result could be applied. The same will happen with any mixed problem.
 Again, hypothesis (H2) is not necessary in this case and we would need to
add the hypothesis that $f_0=\infty$ and $f^\infty=0$.
 \end{remark}


 \section{Periodic boundary value problem with constant potential} \label{sect_ex_per}

 This section is devoted to the particular case in which the potential $a$
is constant. As we will see, in this situation it is possible to calculate
the exact value of $\gamma$.

 It is  well known (see \cite{cabada2, zettl}) that the eigenvalues associated
 to the periodic problem
 \begin{equation}\label{per-const}
 u''+\lambda\, u=0, \quad u(0)=u(T),\;u'(0)=u'(T)
 \end{equation}
 are $\lambda_n=(2n\pi/T)^2$ with $n=0,1,2,\dots$
The eigenfunctions associated to the first eigenvalue $\lambda_P=0$ are
the constants, which can be written as multiples of a representative
eigenfunction $v_P(t)\equiv1$.

 Moreover, the related Green's function is strictly negative in the square
$I \times I$ if and only if $\lambda<0$ and it is nonnegative on $I \times I$
if and only if $0<\lambda \le (\pi/T)^2$ (see \cite{cacilu} for details).

For $\lambda=\rho^2$ a nonresonant value, the explicit form of $G_P$ is
the following (see \cite{cabada1, cabada2, ma,ZA}):
 \[
 G_P(t,s)=\begin{cases}
 \frac{\sin \rho(t-s)+\sin \rho(T-t+s)}{2\rho(1-\cos \rho T)},&
 0\le s\le t\le T, \\[4pt]
 \frac{\sin \rho(s-t)+\sin \rho(T-s+t)}{2\rho(1-\cos \rho T)},&
 0\le t\le s\le T \,.
 \end{cases}
 \]
 From \eqref{int-a} it is clear that
 $$
g(t)=\int_0^T G_P(t,s)\,ds=\frac{1}{\rho^2},
$$
 therefore we define
 \[
 \gamma=\min_{t\in[0,T]}
 \frac{\int_0^TG_P^+(t,s)\,ds}{\int_0^TG_P ^-(t,s)\,ds}>1
 \]
 for all $\rho>\pi/T$, $\rho\neq k\pi/T$, $k=1, 2, \ldots$

 Let us make a careful study of this value $\gamma$.
 It is very well-known that the Green's function related to the
periodic problem \eqref{per-const} satisfies that
 \[
G_P(t,s)=G_P(0,t-s) \quad \text{and} \quad G_P(t,s)=G_P(T-t,T-s)
\]
 (see \cite{cabada2} for the details). Therefore,
 \begin{equation*}
 \int_0^T G_P(t,s)\,ds= \int_0^t G_P(t,s)\,ds + \int_t^T G_P(t,s)\,ds,
 \end{equation*}
 where
 \begin{equation*}
 \int_0^t G_P(t,s)\,ds= \int_0^t G_P(0,t-s)\,ds = \int_0^t G_P(0,T+s-t)\,ds
= \int_{T-t}^T G_P(0,s)\,ds
 \end{equation*}
 and
 \begin{equation*}
 \int_t^T G_P(t,s)\,ds= \int_t^T G_P(0,T+s-t)\,ds = \int_T^{2T-t} G_P(0,s)\,ds
= \int_0^{T-t} G_P(0,s)\,ds,
 \end{equation*}
 that is
 \begin{equation*}
 \int_0^T G_P(t,s)\,ds= \int_0^T G_P(0,s)\,ds \quad \forall \, t\in [0,T].
 \end{equation*}
 The same argument is valid for both the positive and the negative parts of
$G_P$, that is
 \begin{equation*}
 \int_0^T G^+_P(t,s)\,ds= \int_0^T G^+_P(0,s)\,ds \quad \text{and} \quad
 \int_0^T G^-_P(t,s)\,ds= \int_0^T G^-_P(0,s)\,ds
 \end{equation*}
for all $t\in [0,T]$,
so the ratio $\frac{\int_0^T G_P^+(t,s) \,ds}{\int_0^T G_P^-(t,s) \,ds}$
is constant for all $t\in[0,T]$.

 This implies that we can restrict our analysis to the case $t=0$, that is,
to assume that
 \[
 \gamma=
 \frac{\int_0^TG_P^+(0,s)\,ds}{\int_0^TG_P^-(0,s)\,ds}.
 \]
 We have that
 \[
 G_P(0,s)=\frac{\sin \rho s+ \sin \rho(T-s)}{2\rho(1-\cos\rho T)},
 \]
 so $G_P(0,s)=0$ if and only if $s=\frac{T}{2}+\frac{(2k+1)\pi}{2\rho}$.
We will consider four cases:
 \begin{itemize}
 \item[Case 1A:]  $G_P(0,\frac{T}{2})\,G_P(0,0)>0$ and $G_P(0,\frac{T}{2})>0$;
 \item[Case 1B:]  $G_P(0,\frac{T}{2})\,G_P(0,0)>0$ and $G_P(0,\frac{T}{2})<0$;
 \item[Case 2A:]  $G_P(0,\frac{T}{2})\,G_P(0,0)<0$ and $G_P(0,\frac{T}{2})>0$;
 \item[Case 2B:]  $G_P(0,\frac{T}{2})\,G_P(0,0)<0$ and $G_P(0,\frac{T}{2})<0$.
 \end{itemize}
Computing these values, we find that \begin{itemize}
 \item[] if $\frac{(4k+1)\pi}{T}<\rho<\frac{(4k+2)\pi}{T}$ for some
$k\in\mathbb{N}_0$, we are in case 2A and $\gamma=\frac{2k+1}{2k+1-\sin(\rho T/2)}$;
 \item[] if $\frac{(4k+2)\pi}{T}<\rho<\frac{(4k+3)\pi}{T}$ for some $k\in\mathbb{N}_0$,
 we are in case 2B and $\gamma=\frac{2k+1-\sin(\rho T/2)}{2k+1}$;
 \item[] if $\frac{(4k-1)\pi}{T}<\rho<\frac{4k\pi}{T}$ for some $k\in\mathbb{N}$,
we are in case 1B and $\gamma=\frac{2k}{2k+\sin(\rho T/2)}$;
 \item[] if $\frac{4k\pi}{T}<\rho<\frac{(4k+1)\pi}{T}$ for some $k\in\mathbb{N}$,
 we are in case 1A and $\gamma=\frac{2k+\sin(\rho T/2)}{2k}$.
 \end{itemize}
 In the cases where $\rho=(2k+1)\frac{\pi}{T}$ for some $k\in\mathbb{N}$, the value of
 $\gamma$ coincides with the limit when $\rho\to (2k+1)\frac{\pi}{T}$.
The graph of $\gamma$ for a given value $\rho$ is sketched in
Figure \ref{gamma_P}.
 \begin{figure}[ht]
 \begin{center}
\includegraphics[width=0.6\textwidth]{fig1} 
\end{center}
 \caption{Graph of $\gamma$ for the periodic problem.}\label{gamma_P}
 \end{figure}


 \section{Dirichlet boundary value problem with constant potential}
 \label{sect-Dir-constant}

 Let us now try to prove some analogue results for the Dirichlet boundary
conditions. In this case, the eigenvalues for the Dirichlet problem
 \[
 u''(t)+\lambda\, u(t)=0,  \text{ for } t\in(0,T), \quad u(0)=u(T)=0,
 \]
 are $\lambda_n=(n\pi/T)^2$ for $n=1,2,3\dots$, and it follows easily
that the eigenfunctions associated to $\lambda_D\equiv \lambda_1=(\pi/T)^2$
are the multiples of the function $v_D(t)=\sin (\frac{\pi t}{T})$.

 It is  well known that the associated Green's function is strictly negative
if and only if $\lambda<\lambda_1=(\pi/T)^2$, and it changes sign for
 any nonresonant value of $\lambda>(\pi/T)^2$.

Considering $\lambda=\rho^2$
 for $\rho\neq \frac{n\pi}{T}$, with $n\in\mathbb{N}$, we have
 $\int_0^T G_D(t,s)\sin(\frac{\pi s}{T})\,ds>0$ for $t\in(0,T)$, and we define
 \[
 \gamma(\rho)=\inf_{t\in(0,T)}\gamma(t,\rho)=
 \inf_{t\in(0,T)}
 \frac{\int_0^TG_D^+(t,s)\sin(\frac{\pi s}{T})\,ds}{\int_0^T G_D
 ^-(t,s)\sin(\frac{\pi s}{T})\,ds}.
 \]
 The explicit formula for the Green's function in the nonresonant cases
is given by (see \cite{cabada2})
 \begin{equation}
 \label{e-G-Dir}
 G_D(t,s)=\begin{cases}
 G_1(t,s)=-\frac{\sin (\rho s)\sin \rho(T-t)}{\rho\sin (\rho T)},&
 0\le s\le t\le T, \\[4pt]
 G_2(t,s)=-\frac{\sin (\rho t)\sin \rho(T-s)}{\rho\sin (\rho T)},&
 0\le t\le s\le T \,.
 \end{cases}
 \end{equation}

 We will consider two cases:
\begin{itemize}
 \item[Case 1:] $\frac{(2n-1)\pi}{T}<\rho<\frac{2n\pi}{T}$ for $n\in \mathbb{N}$;
 \item[Case 2:] $\frac{2n\pi}{T}<\rho<\frac{(2n+1)\pi}{T}$ for $n\in \mathbb{N}$.
 \end{itemize}
 In case~1 the function $\gamma(t,\rho)$ has a different computation
in each of the $4n-1$ intervals
\begin{gather*}
 \big]0,T-\frac{(2n-1)\pi}{\rho T}\big], \quad
 \big[T-\frac{(2n-1)\pi}{\rho T},\frac{\pi}{\rho T}\big], \quad
 \big[\frac{\pi}{\rho T},T-\frac{(2n-2)\pi}{\rho T}\big], \\
 \big[T-\frac{(2n-2)\pi}{\rho T},\frac{2 \pi}{\rho T}\big],\quad \dots \quad
 \big[\frac{(2n-2)\pi}{\rho T},T-\frac{\pi}{\rho T}\big], \\
\big[T-\frac{\pi}{\rho T},\frac{(2n-1)\pi}{\rho T}\big],\quad
 \big[\frac{(2n-1)\pi}{\rho T},T\big[
\end{gather*}
and in case 2, it has a different computation in each of the $4n+1$ intervals
 \[
 \big]0,T-\frac{2n\pi}{\rho T}\big],\quad
 \big[T-\frac{2n\pi}{\rho T},\frac{\pi}{\rho T}\big], \;\dots,\;
 \big[T-\frac{\pi}{\rho T},\frac{2n\pi}{\rho T}\big],\,
 \big[\frac{2n\pi}{\rho T},T\big[\,.
 \]
 In both cases, given a fixed $\rho$ it is easy to calculate the value of
$\gamma(t,\rho)$. However the general expression for an arbitrary $\rho$
requires very long computations which are not fundamental for the purpose
 of this paper. Because of this, we are going to calculate the general
expression of $\gamma(\rho)$ only for the first intervals of $\rho$,
in particular for $\rho<6\pi/T$.

 For $\rho<\frac{6 \pi}{T}$, we can see that the infimum is attained at $t=0$,
so we will restrain our analysis to the first interval of $t$ in both cases
in order to obtain the exact expression of $\gamma(\rho)$ for
$\rho<6\pi/T$.

 In case~1 we have
 \begin{align*}
&\int_0^T G_D^+(t,s)\sin\big(\frac{\pi s}{T}\big)\,ds \\
&= \int_{T-\frac{\pi}{\rho}}^T G_2(t,s)\sin\big(\frac{\pi s}{T}\big)\,ds
+ \sum_{i=2}^n\int_{T-\frac{(2i-1)\pi}{\rho T}}^{T-\frac{(2i-2)\pi}{\rho T}}
 G_2(t,s)\sin\big(\frac{\pi s}{T}\big)\,ds
 \end{align*}
 and
 \begin{align*}
&-\int_0^T G_D^-(t,s)\sin\big(\frac{\pi s}{T}\big)\,ds \\
& =\int_0^t G_1(t,s)\sin\big(\frac{\pi s}{T}\big)\,ds
+\int_t^{T-\frac{(2n-1)\pi}{\rho T}} G_2(t,s)\sin\big(\frac{\pi s}{T}\big)\,ds\\
&\quad + \sum_{i=1}^{n-1} \int_{T-\frac{2i\pi}{\rho T}}^{T-\frac{(2i-1)\pi}{\rho T}}
 G_2(t,s)\sin\big(\frac{\pi s}{T}\big)\,ds \\
&=\frac{\sin\big(\frac{\pi t}{T}\big)}{\rho^2-\big(\frac{\pi}{T}\big)^2}-
 \int_0^T G_D^+(t,s)\sin\big(\frac{\pi s}{T}\big)\,ds
 \end{align*}
 so
 \begin{align*}
 \gamma(t,\rho)
&=\Big(\int_{T-\frac{\pi}{\rho T}}^T G_2(t,s)\sin\big(\frac{\pi s}{T}\big)\,ds+
 \sum_{i=2}^n \int_{T-\frac{(2i-1)\pi}{\rho T}}^{T-\frac{(2i-2)\pi}{\rho T}}
  G_2(t,s) \sin\big(\frac{\pi s}{T}\big)\,ds\Big) \\
&\quad\div \Big(\int_{T-\frac{\pi}{\rho T}}^T G_2(t,s)\sin\big(\frac{\pi s}{T}\big)\,ds
 + \sum_{i=2}^n\int_{T-\frac{(2i-1)\pi}{\rho T}}^{T-\frac{(2i-2)\pi}{\rho T}}
 G_2(t,s)\sin\big(\frac{\pi s}{T}\big)\,ds \\
&\quad  -\frac{\sin\big(\frac{\pi t}{T}\big)}{\rho^2-\big(\frac{\pi}{T}\big)^2}
 \Big).
 \end{align*}
 Doing a similar study for case~2 we get
 \[
 \gamma(t,\rho)
=\frac{
 \sum_{i=1}^n\int_{T-\frac{2i\pi}{\rho T}}^{T-\frac{(2i-1)\pi}{\rho T}}
 G_2(t,s)\sin\big(\frac{\pi s}{T}\big)\,ds}
 {\sum_{i=1}^n\int_{T-\frac{2i\pi}{\rho T}}^{T-\frac{(2i-1)\pi}{\rho T}}
G_2(t,s)\sin\big(\frac{\pi s}{T}\big)\,ds
 -\frac{\sin\big(\frac{\pi t}{T}\big)}{\rho^2-\big(\frac{\pi}{T}\big)^2}}.
 \]
Using the previous expressions it is immediate to calculate $\gamma(t,\rho)$
for any fixed value of $\rho$ and $T$. For instance, computing $\gamma(t,\rho)$
for $T=1$ we obtain:
\\
If $\rho \in (\pi,2 \pi)$, then
\[
 \gamma(t,\rho)=\frac{\sin{\rho\,t}\sin\frac{\pi^2}{\rho}} {\sin{\rho\,t}
 \sin\frac{\pi^2}{\rho} + \sin{\rho}  \sin{\pi\,t}};
\]
If $\rho \in (2 \pi,3 \pi)$, then
\[
\gamma(t,\rho)=\frac{\sin\rho\,t\left(\sin\frac{\pi^2}{\rho}
+  \sin\frac{2 \pi^2}{\rho}\right)} {\sin\rho t\left(\sin\frac{\pi^2}{\rho}
 +  \sin\frac{2 \pi^2}{\rho}\right) -  \sin\rho\sin\pi t};
\]
If $\rho \in (3 \pi,4 \pi)$, then
\[
\gamma(t,\rho)=\frac{\sin\rho t \left(\sin\frac{\pi^2}{\rho}
+  \sin\frac{2 \pi^2}{\rho}  +  \sin\frac{3 \pi^2}{\rho}\right) }
{\sin\rho\,t\left(\sin\frac{\pi^2}{\rho}  +  \sin\frac{2 \pi^2}{\rho}
 +  \sin\frac{3 \pi^2}{\rho}\right) +  \sin\rho \sin\pi t};
\]
If $\rho \in (4 \pi,5 \pi)$, then
\[
\gamma(t,\rho)=\frac{\sin\rho t\left(\sin\frac{\pi^2}{\rho}
+  \sin\frac{2 \pi^2}{\rho}  +  \sin\frac{3 \pi^2}{\rho}
 +  \sin\frac{4 \pi^2}{\rho} \right)}{\sin\rho t \left(\sin\frac{\pi^2}{\rho}
 +  \sin\frac{2 \pi^2}{\rho}  +  \sin\frac{3 \pi^2}{\rho}
+ \sin\frac{4 \pi^2}{\rho} \right)-\sin\rho  \sin\pi t} ;
\]
If $\rho \in (5 \pi,6 \pi)$, then
\begin{align*}
\gamma(t,\rho)
&=\Big(\sin\rho t  \Big(\sin\frac{2 \pi^2}{\rho}
+  \sin\frac{3 \pi^2}{\rho} +  \sin\frac{4 \pi^2}{\rho}
+  \sin\frac{5 \pi^2}{\rho}\Big) + 2\Big(1-\frac{\pi^2}{\rho^2}\Big)
\sin\rho t \Big)\\
&\quad\div\Big(\sin\rho t \Big(\sin\frac{2 \pi^2}{\rho}
+  \sin\frac{3 \pi^2}{\rho} +  \sin\frac{4 \pi^2}{\rho}
+ \sin\frac{5 \pi^2}{\rho}\Big)\\
&\quad  + \sin\rho\sin\pi t
+ 2\Big(1-\frac{\pi^2}{\rho^2}\Big)\sin\rho\,t\Big).
 \end{align*}

 In Figure \ref{gamma_t} we have a sketch of the function $\gamma(t,10.8)$
for $T=1$.

 \begin{figure}[ht]
 \begin{center}
\includegraphics[width=0.6\textwidth]{fig2}
\end{center}
 \caption{Graph of $\gamma(t,10.8)$ for the Dirichlet problem.}\label{gamma_t}
 \end{figure}


 Computing the limit
 \[
 \gamma(\rho)=\lim_{t\to 0}\gamma(t,\rho),
 \]
 we get the following expressions for $\gamma(\rho)$:\\
If $\rho \in (\pi,2 \pi)$, then
\[
\gamma(\rho)=1-\frac{\pi\sin{\rho}}{\pi\sin{\rho}
+ \rho \sin\frac{\pi^2}{\rho}};
\]
If $\rho \in (2 \pi,3 \pi)$, then
\[
\gamma(\rho)=1+\frac{\pi\, \sin\rho}{-\pi\sin\rho
+ \rho\left(\sin\frac{\pi^2}{\rho} + \sin\frac{2 \pi^2}{\rho}\right)};
\]
If $\rho \in (3 \pi,4 \pi)$, then
\[
\gamma(\rho)=1-\frac{\pi\, \sin\rho}{\pi\sin\rho
+ \rho\left(\sin\frac{\pi^2}{\rho} + \sin\frac{2 \pi^2}{\rho}
 + \sin\frac{3 \pi^2}{\rho}\right)} ;
\]
If $\rho \in (4 \pi,5 \pi)$, then 
\[
\gamma(\rho)=1+\frac{\pi\, \sin\rho}{-\pi\sin\rho
 + \rho\left(\sin\frac{\pi^2}{\rho} + \sin\frac{2 \pi^2}{\rho}
+ \sin\frac{3 \pi^2}{\rho}+ \sin\frac{4 \pi^2}{\rho}\right)} ;
\]
If $\rho \in (5 \pi,6 \pi)$, then 
\begin{align*}
\gamma(\rho)&=1 -\frac{\pi\, \sin\rho}{\pi\sin\rho
+ \rho \left(\sin\frac{\pi^2}{\rho} + \sin\frac{2 \pi^2}{\rho}
+ \sin\frac{3 \pi^2}{\rho}+ \sin\frac{4 \pi^2}{\rho}
+ \sin\frac{5 \pi^2}{\rho}\right) +2\frac{\rho^2-\pi^2}{\rho}}.
 \end{align*}
 Graphically the function $\gamma(\rho)$ is represented in Figure
\ref{gamma_D} for $T=1$.

 \begin{figure}[ht]
 \begin{center}
\includegraphics[width=0.6\textwidth]{fig3}
\end{center}
 \caption{Graph of $\gamma$ for the Dirichlet problem.}\label{gamma_D}
 \end{figure}

Let us now see some examples.

 \begin{example} \rm
 The Dirichlet BVP
 \begin{equation}\label{ex-D-1}
 u''(t)+60 u(t)=t(1-t) ,  \text{ for } t\in(0,1)\quad
 u(0)=u(1)=0
 \end{equation}
 has a positive solution, since $\gamma(\sqrt{60})\approx 1.36>4/3$
 and $\frac{3\sin(\pi t)}{4\pi}\le t(1-t)\le \frac{\sin(\pi t)}{\pi}$,
 but the solution of the Dirichlet BVP
 \begin{equation}\label{ex-D-2}
 u''(t)+60 u(t)=t ,  \text{ for } t\in(0,1)\quad
 u(0)=u(1)=0
 \end{equation}
 changes sign. We can see the respective solutions in Figures
\ref{fig-ex-D-1} and \ref{fig-ex-D-2}.
\end{example}

 \begin{figure}[ht]
 \begin{center}
\includegraphics[width=0.6\textwidth]{fig4}
\end{center}
 \caption{Solution of problem \eqref{ex-D-1}}.
 \label{fig-ex-D-1}
 \end{figure}

 \begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig5}
 \end{center}
 \caption{Solution of problem \eqref{ex-D-2}.} \label{fig-ex-D-2}
 \end{figure}


\begin{remark} \rm
Analogous arguments and calculations can be done for the Neumann
and mixed problems.
\end{remark}

\subsection*{Acknowledgments}
A. Cabada and L. L\'opez-Somoza were partially supported by Ministerio
de Econom\'ia y Competitividad, Spain, and FEDER, project MTM2013-43014-P,
and by the Agencia Estatal de Investigaci\'on (AEI) of Spain under grant
MTM2016-75140-P, co-financed by the European Community fund FEDER.

L. L\'opez-Somoza was spartially supported by 
FPU scholarship, Ministerio de Educaci\'on,
Cultura y Deporte, Spain.

R. Engui\c{c}a was partially supported by Funda\c{c}ao para a Ci\^{e}ncia e a
Tecnologia, Portugal, UID/MAT/04561/2013.


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