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\markboth{\hfil Remarks on semilinear problems \hfil
EJDE--2003/18} {EJDE--2003/18\hfil Jose Mar\'{\i}a Almira \& Naira
Del Toro \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2003}(2003), No. 18, pp. 1--11. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Remarks on semilinear problems with nonlinearities depending on
  the derivative
 %
\thanks{ {\em Mathematics Subject Classifications:} 34B15, 34L30.
\hfil\break\indent
{\em Key words:} Nonlinear boundary-value problem, Neumann and Periodic problems.
\hfil\break\indent
\copyright 2003 Southwest Texas State University. \hfil\break\indent
Submitted December 5, 2002. Published February 20, 2003.
\hfil\break\indent
Partially supported by grant FQM-0178 from the Junta de Andaluc\'{\i}a,
\hfil\break\indent
 Grupo de Investigaci\'{o}n ``Aproximaci\'{o}n y M\'{e}todos
 Num\'{e}ricos'' .} }
\date{}
%
\author{Jose Mar\'{\i}a Almira \& Naira Del Toro}
\maketitle

\begin{abstract}
 In this paper, we continue some work by
 Ca\~{n}ada and Dr\'{a}bek \cite{C-D}  and Mawhin \cite{M}  on the
 range of the  Neumann and Periodic boundary value problems:
 \begin{gather*}
 \mathbf{u}''(t)+\mathbf{g}(t,\mathbf{u}'(t))=
 \overline{\mathbf{f}}+\widetilde{\mathbf{f}}(t), \quad t\in (a,b) \\
 \mathbf{u}'(a)=\mathbf{u}'(b)=0 \\
 \text{or}\quad \mathbf{u}(a)=\mathbf{u}(b),\quad \mathbf{u}'(a)=\mathbf{u}'(b)
 \end{gather*}
 where $\mathbf{g}\in C([a,b]\times \mathbb{R}^{n},\mathbb{R}^{n})$,
 $\overline{\mathbf{f}}\in \mathbb{R}^{n}$, and
 $\widetilde{\mathbf{f}}$ has mean value zero. For the Neumann problem
 with $n>1$, we prove that for a fixed $\widetilde{\mathbf{f}}$ the range can
 contain an infinity continuum. For the one dimensional case,
 we study the asymptotic  behavior of the range in both problems.
\end{abstract}

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\section{Introduction}

Let us consider the resonance problem
\begin{equation}
\begin{gathered}
u''(t)+g(u'(t))=f(t), \quad t\in (a,b) \\
u'(a)=u'(b)=0
\end{gathered} \label{problema}
\end{equation}
where $f\in C[a,b]$ and $g:\mathbb{R}\to \mathbb{R}$ is
continuous. The linearized part of (\ref{problema}) is the
resonance system
\begin{equation}
\begin{gathered}
u''(t)=f(t), \quad t\in (a,b) \\
u'(a)=u'(b)=0
\end{gathered} \label{linear}
\end{equation}
and the corresponding eigenfunction is $u_{1}(t)=1$.  The change
of variable $v=u'$ transforms (\ref{linear})  into the problem
\begin{equation}
\begin{gathered}
v'(t)=f(t), \quad t\in (a,b) \\
v(a)=v(b)=0
\end{gathered}  \label{p2}
\end{equation}
which obviously is solvable if and only if $\int_{a}^{b}f(t)dt=0$.
Moreover, its solution is given by $v(t)=\int_{a}^{t}f(s)ds$.
Hence (\ref{linear}) is
solvable if and only if $f=\widetilde{f}\in \widetilde{C}[a,b]:=\{\widetilde{%
f}\in C[a,b]:\int_{a}^{b}\widetilde{f}(t)dt=0\}$ and its set of
solutions is
\begin{equation*}
u_{c}(t)=c+\int_{a}^{t}v(s)ds
\end{equation*}
where $c\in \mathbb{R}$ and $v(t)=\int_{a}^{t}f(s)ds$.
Let us now consider  problem (\ref{problema}). When we decompose
\begin{equation}
f(t)=s+\widetilde{f}(t)  \label{descom}
\end{equation}
where $s\in \mathbb{R}$ and $\widetilde{f}\in \widetilde{C}[a,b]$,
it is quite natural to ask for which values $s\in \mathbb{R}$
the problem (\ref {problema}) is solvable. This question has been
studied by several authors.
In particular, Ca\~{n}ada and Dr\'{a}bek (see \cite{C-D}) proved
 that if $g\in C^{1}(\mathbb{R})$ and is bounded, then for each
$\widetilde{f}$ there is a unique value $s=s(\widetilde{f})\in
\mathbb{R}$ such that (\ref {problema}) is solvable. Moreover, in
such a case they also proved that the
map $s(\cdot ):\widetilde{C}[a,b]\to \mathbb{R}$ , $\widetilde{f}%
\to s(\widetilde{f})$ is continuously differentiable and satisfies $%
|s(\widetilde{f})|\leq \|g\|$ for all $\widetilde{f}\in
\widetilde{C}[a,b]$, where $\|g\|= \sup_{t \in \mathbb{R}}|g(t)|$.
In the same paper the authors noted that their proofs are also
applicable to the more general problem
\begin{equation}
\begin{gathered}
u''(t)+g(t,u'(t))=s+\widetilde{f}(t), \quad t\in (a,b) \\
u'(a)=u'(b)=0
\end{gathered} \label{canadanuevo}
\end{equation}
(with $g\in C^{1}([a,b]\times \mathbb{R},\mathbb{R})$ and bounded)
and also to the periodic problem
\begin{equation}
\begin{gathered}
u''(t)+g(t,u'(t))=s+\widetilde{f}(t), \quad t\in (a,b)\\
u(a)=u(b),\quad u'(a)=u'(b);
\end{gathered} \label{canadanuevo2}
\end{equation}
and proposed as an open question to study these kind of problems
for systems of equations and for higher order equations. This was
made by Mawhin in \cite {M}. In particular, he studied the
problems
\begin{equation}
\begin{gathered}
\mathbf{u}''(t)+\mathbf{g}(t,\mathbf{u}'(t))=
\overline{\mathbf{f}}+\widetilde{\mathbf{f}}(t), \quad t\in (a,b) \\
\mathbf{u}'(a)=\mathbf{u}'(b)=\mathbf{0}
\end{gathered}  \label{problemaw1}
\end{equation}
and
\begin{equation}
\begin{gathered}
\mathbf{u}''(t)+\mathbf{g}(t,\mathbf{u}'(t))=
\overline{\mathbf{f}}+\widetilde{\mathbf{f}}(t), \quad t\in (a,b) \\
\mathbf{u}(a)=\mathbf{u}(b),\quad \mathbf{u}'(a)=\mathbf{u}'(b),
\end{gathered} \label{problemaw2}
\end{equation}
where $\mathbf{g}:[a,b]\times \mathbb{R}^{n}\to
\mathbb{R}^{n}$ is a
Carath\'{e}dory function, $\mathbf{u:}[a,b]\to \mathbb{R}^{n}$,
$\overline{\mathbf{f}}\in \mathbb{R}^{n}$ and
\begin{equation*}
\widetilde{\mathbf{f}}\in \widetilde{L^{1}}([a,b],\mathbb{R}^{n}):=\{%
\widetilde{\mathbf{f}}\in L^{1}([a,b],\mathbb{R}^{n}):\int_{a}^{b}\widetilde{%
\mathbf{f}}(t)dt=0\};
\end{equation*}
and proved that if
\begin{equation}
\lim_{\|\mathbf{v\|}_{2}\to \infty }\big\| \mathbf{g}(t,\mathbf{v})/
\|\mathbf{v}\|_{2} \big\| _{2}=0 \quad\text{uniformly a.e. in }
t\in [a,b],  \label{condicionmawhin}
\end{equation}
then for each $\widetilde{\mathbf{f}}\in
\widetilde{L^{1}}([a,b],\mathbb{R}^{n})$ the sets
\begin{gather*}
\mathcal{J}_{\widetilde{\mathbf{f}}}^{(\mathcal{N})} =\{\overline{\mathbf{f%
}}\in \mathbb{R}^{n}:\text{the problem }(\ref{problemaw1})\text{ is solvable}%
\} \\
\mathcal{J}_{\widetilde{\mathbf{f}}}^{(\mathcal{P})} =\{\overline{\mathbf{f%
}}\in \mathbb{R}^{n}:\text{the problem }(\ref{problemaw2})\text{ is solvable}%
\}
\end{gather*}
are both nonempty, where $\| \cdot\|_{2}$ denotes the Euclidean
norm of $\mathbb{R}^n$.
Moreover, he also proved that for $n=1$ and $\widetilde{f}%
\in \widetilde{L^{1}}(a,b):=\widetilde{L^1}([a,b],\mathbb{R})$,
$\#\mathcal{J}_{\widetilde{f}}^{(\mathcal{N})}
=\#\mathcal{J}_{\widetilde{f}}^{(\mathcal{P})}=1$ and stated the
uniqueness problem for $n>1$ as an open question. In this note we
solve this problem in the negative sense for the Neumann case
(\ref{problemaw1}).

For $n=1$ and $g\in C([a,b]\times \mathbb{R},\mathbb{R})$
satisfying (\ref {condicionmawhin}), we denote by
$s_{\mathcal{N}}(\widetilde{f})$ the unique
element of $\mathcal{J}_{\widetilde{f}}^{(\mathcal{N})}$ and by $s_{\mathcal{%
P}}(\widetilde{f})$ the unique element of $\mathcal{J}_{\widetilde{f}}^{(%
\mathcal{P})}$. We study the asymptotic
behavior of the functionals $s_{\mathcal{N}}(\widetilde{f})$ and $s_{%
\mathcal{P}}(\widetilde{f})$ for $\|\widetilde{f}\|\to
\infty $  when the uniqueness results are applicable.

\section{Uniqueness Problem}

The first contribution of this note to the subject is that we
solve for the Neumann problem (\ref{problemaw1}) the uniqueness
question in the negative
sense for all $n>1$. With this objective in mind, we take $h:\mathbb{%
R\to R}$ a $C^{\infty }$ function such that it is bounded
and
satisfies $h(x)=x$ for all $x\in [ -2,2]$ and we set $\widetilde{%
\mathbf{f}}=\mathbf{0}$ and
\begin{equation*}
\mathbf{g(}t,x_{1},x_{2},x_{3},\dots ,x_{n})=-(-h(x_{2}),h(x_{1}),0,\dots ,0).
\end{equation*}
Then $\mathbf{g}:[0,2\pi ]\times \mathbb{R}^{n}\to
\mathbb{R}^{n}$  belongs to $C^{\infty }([0,2\pi ]\times
\mathbb{R}^{n},\mathbb{R}^{n})$ and it is bounded. Let us now
consider the problem
\begin{equation}
\begin{gathered}
\mathbf{u}''(t)+\mathbf{g}(t,\mathbf{u}'(t))=%
\overline{\mathbf{f}}, \quad  t\in (0,2\pi ) \\
\mathbf{u}'(0)=\mathbf{u}'(2\pi )=\mathbf{0}
\end{gathered}  \label{contraejemplo}
\end{equation}
and let $\alpha \in [ -1,1]$ be fixed. We set
$\mathbf{u}_{\alpha}(t)=(\alpha \sin (t-\frac{\pi }{2}),
\alpha t-\alpha \cos (t-\frac{\pi }{2}),0,\dots ,0)$ with
$\alpha \in [ -1,1]$. Then
$\mathbf{u}_{\alpha }\in C^{2}([0,2\pi ],\mathbb{R}^{n})$ and
$\mathbf{u}_{\alpha }'(t)=(\alpha \cos (t-\frac{\pi }{2}),\alpha
+\alpha \sin (t-\frac{\pi }{2}),0,\dots ,0)$, so that
$\mathbf{u}_{\alpha }'(0)=\mathbf{u}_{\alpha }'(2\pi )=\mathbf{0}$
and
\begin{eqnarray*}
\mathbf{u}_{\alpha }''(t) &=&(-\alpha \sin (t-\frac{\pi }{2}%
),\alpha \cos (t-\frac{\pi }{2}),0,\dots ,0) \\
&=&(-(\alpha +\alpha \sin (t-\frac{\pi }{2})),\alpha \cos (t-\frac{\pi }{2}%
),0,\dots ,0)+(\alpha ,0,0,\dots ,0) \\
&=&-\mathbf{g(}t\mathbf{,u}_{\alpha }'(t)\mathbf{)}+(\alpha
,0,0,\dots ,0)
\end{eqnarray*}
Hence $\mathbf{u}_{\alpha }$ solves (\ref{contraejemplo}) with $\overline{%
\mathbf{f}}=(\alpha ,0,\dots ,0)$ and we have proved that there
exists a continuum of vectors $\overline{\mathbf{f}}\in
\mathbb{R}^{n}$ for which the problem (\ref{contraejemplo}) is
solvable. Moreover, we have got such a
result not only for $\mathbf{g}(t,\mathbf{x})$ continuous but also $%
C^{\infty }$ and bounded, so that $\mathbf{g}(t,\mathbf{x})$
satisfies the hypothesis of the existence and uniqueness results
in the papers by Mawhin (see \cite[Theorems 1 and 3]{M}) and
Ca\~{n}ada and Dr\'{a}bek (see \cite[Theorem 3.3]{C-D}). This
proves that the mentioned uniqueness result for $n=1$ is
impossible to generalize to higher dimensions. Of course, the same
problem is still open for the periodic case.

\section{Asymptotic behavior}

In this section we set $n=1$ and we consider the problems (\ref{canadanuevo})
and (\ref{canadanuevo2}). Moreover, in order to have existence
of
solutions, we assume that $g(t,u)$ satisfies that $\lim_{|u\mathbf{|}%
\to \infty }\frac{g(t,u)}{|u|}=0$ uniformly in $t\in [ a,b]$.
With these hypotheses at hands we know that for each
$\widetilde{f}\in \widetilde{C}[a,b]$, $\mathcal{J}_{\widetilde{f}}^{(%
\mathcal{N})}=\{s_{\mathcal{N}}(\widetilde{f})\}$ and $\mathcal{J}_{%
\widetilde{f}}^{(\mathcal{P})}=\{s_{\mathcal{P}}(\widetilde{f})\}$, where $%
s_{\mathcal{N}}:\widetilde{C}[a,b]\to \mathbb{R}$ and $s_{\mathcal{P}%
}:\widetilde{C}[a,b]\to \mathbb{R}$ are certain
functionals. Furthermore, the change of variables $v=u'$
transforms (\ref {canadanuevo}) and (\ref{canadanuevo2}) into the
problems
\begin{equation}
\begin{gathered}
v'(t)+g(t,v(t))=s+\widetilde{f}(t), \quad t\in (a,b) \\
v(a)=v(b)=0
\end{gathered}  \label{problematransformado}
\end{equation}
and
\begin{equation}
\begin{gathered}
v'(t)+g(t,v(t))=s+\widetilde{f}(t), \quad t\in (a,b) \\
v(a)=v(b), \quad \int_{a}^{b}v(t)dt=0.
\end{gathered}   \label{problematransformado2}
\end{equation}
Thus, if $w(t)$ solves (\ref{problematransformado}) and $\omega (t)$
solves (\ref{problematransformado2}) and we integrate between $a$ and $b$
both sides of the equation, we get
\begin{equation*}
s_{\mathcal{N}}(\widetilde{f})=\frac{1}{b-a}\int_{a}^{b}g(t,w(t))dt
\quad \text{and}\quad
s_{\mathcal{P}}(\widetilde{f})=\frac{1}{b-a}\int_{a}^{b}g(t,\omega(t))dt.
\end{equation*}
We will use the formulas above in order to prove certain
asymptotic results for the functionals $s_{\mathcal{N}}(\cdot )$
and $s_{\mathcal{P}}(\cdot )$.

Now we state and prove the main results of this section.

\begin{theorem} \label{thm1}
Let us set $\Theta =\{\frac{1}{b-a}\int_{a}^{b}g(t,v_{0})dt:v_{0}\in \mathbb{%
R}\}$. Then for each $g_{0}\in \overline{\Theta }$, the closure of
$\Theta $
in $\mathbb{R}$, there exists a sequence $\{\widetilde{f}_{n}\}_{n=1}^{%
\infty }\subset \widetilde{C}[a,b]$ such that $\lim_{n\to \infty }\|%
\widetilde{f}_{n}\|=\infty $ and $\lim_{n\to \infty }s_{\mathcal{N}}(%
\widetilde{f}_{n})=g_{0}$, where $\|\widetilde{f}_{n}\|= \sup_{t
\in [a,b]}|\widetilde{f}_{n}(t)|$.
\end{theorem}


\paragraph{Proof} Let
$g_{0}=\frac{1}{b-a}\int_{a}^{b}g(t,v_{0})dt\in \Theta $
be arbitrarily chosen. We define for each $n>2(b-a)^{-1}$ a function
$w_{n}:[a,b]\to \mathbb{R}$ which satisfies the following
conditions

\begin{itemize}
\item[$a)$]  $w_{n}\in C^{1}[a,b]$

\item[$b)$]  $w_{n}(a)=w_{n}(b)=0$

\item[$c)$]  $w_{n}(a+\frac{1}{2n})=w_{n}(b-\frac{1}{2n})=1$

\item[$d)$]  $w_{n}(t)=v_{0}$ for all $t\in [ a+\frac{1}{n},b
-\frac{1}{n}]$

\item[$e)$]  $\|w_{n}\|\leq |v_{0}|+2$
\end{itemize}
and we set
\begin{equation*}
\widetilde{f}_{n}(t):=w_{n}'(t)+g(t,w_{n}(t))-\frac{1}{b-a}
\int_{a}^{b}g(t,w_{n}(t))dt.
\end{equation*}
It is clear that $a)$ implies that $\widetilde{f}_{n}\in C([a,b])$ for all $%
n\in \mathbb{N}$ and $b)$ implies that $\int_{a}^{b}\widetilde{f}%
_{n}(t)dt=0$. Moreover, using that \ $K=[a,b]\times [
-|v_{0}|-2,|v_{0}|+2]$ is compact and $\{(t,w_{n}(t)):t\in [
a,b]\}\subset K$ for all $n\in \mathbb{N}$, we have that the functions
$g(t,w_{n}(t))$ are uniformly bounded in $[a,b]$, so that the
conditions $b)$ and $c)$ imply that $\lim_{n\to \infty
}\|\widetilde{f}_{n}\|=\infty$.

Then $w=w_{n}$ solves the problem
\begin{gather*}
w'(t)+g(t,w(t))=s_{\mathcal{N}}(\widetilde{f}_{n})+\widetilde{f}
_{n}(t), \quad t\in (a,b) \\
w(a)=w(b)=0
\end{gather*}
with $s_{\mathcal{N}}(\widetilde{f}_{n})=\frac{1}{b-a}\int_{a}^{b}
g(t,w_{n}(t))dt$. We will prove that $\lim_{n\to \infty }s_{%
\mathcal{N}}(\widetilde{f}_{n})=g_{0}$.
In fact, by $d)$ we have that
\begin{align*}
s_{\mathcal{N}}(\widetilde{f}_{n}) &=\frac{1}{b-a}\int_{a}^{b}g(t,w_{n}(t))dt \\
&=\frac{1}{b-a}\Big( \int_{a}^{a+\frac{1}{n}}g(t,w_{n}(t))dt+\int_{a+\frac{%
1}{n}}^{b-\frac{1}{n}}g(t,v_{0})dt+\int_{b-\frac{1}{n}}^{b}g(t,w_{n}(t))dt%
\Big) .
\end{align*}
The uniform boundedness of $g(t,w_{n}(t))$ implies that
\begin{equation*}
\lim_{n\to \infty }\int_{a}^{a+\frac{1}{n}}g(t,w_{n}(t))dt
=\lim_{n\to \infty }\int_{b-\frac{1}{n}}^{b}g(t,w_{n}(t))dt=0.
\end{equation*}
Hence
\begin{equation*}
\lim_{n\to \infty }s_{\mathcal{N}}(\widetilde{f}_{n})=\lim_{n%
\to \infty }\frac{1}{b-a}\int_{a+\frac{1}{n}}^{b-\frac{1}{n}%
}g(t,v_{0})dt=g_{0}
\end{equation*}
which is what we wanted to prove.

Let us now take $g_{0}\in \overline{\Theta }\setminus \Theta $.
Then there exists a sequence of numbers $\{g_{n}\}_{n=1}^{\infty
}\subset \Theta $ and a family of functions
$\{\widetilde{f}_{n,k}\}_{n,k=1}^{\infty }\subset
\widetilde{C}[a,b]$ such that $\|\widetilde{f}_{n,k}\|\geq k$ and
$\big| s_{\mathcal{N}}(\widetilde{f}_{n,k})-g_{n}\big| \leq \frac{1}{k}$
for all $k,n\geq 1$ and $\lim_{n\to \infty }g_{n}=g_{0}$. Thus the sequence
$\{\widetilde{f}_{n,n}\}_{n=1}^{\infty }$ satisfies that
$\lim_{n\to \infty }\|\widetilde{f}_{n,n}\|=\infty $ and
$\lim_{n\to \infty }s_{\mathcal{N}}(\widetilde{f}_{n,n})=g_{0}$.
\hfill$\diamondsuit$

\begin{corollary} \label{coro2}
Let us assume that $g=g(v)\in C(\mathbb{R})$ and $g_{0}\in \overline{g(%
\mathbb{R})}$. Then there exists a sequence $\{\widetilde{f}%
_{n}\}_{n=1}^{\infty }\subset \widetilde{C}[a,b]$ such that
$\lim_{n\to \infty }\|\widetilde{f}_{n}\|=\infty $ and
$\lim_{n\to \infty}s_{\mathcal{N}}(\widetilde{f}_{n})=g_{0}$.
\end{corollary}


\paragraph{Proof} In \cite[Corollary 2]{M} it is shown the
existence of solutions for $n=1$ whenever $g=g(v)$ is continuous.
Hence, it is enough to observe that if $g$ does not depend on the
variable $t$ then $\Theta =g(\mathbb{R})$. \hfill$\diamondsuit$

\begin{theorem} \label{thm3}
Let us assume that $g$ is bounded and set
$\Theta =\{\frac{1}{b-a}\int_{a}^{b}g(t,v_{0})dt:v_{0}\in \mathbb{R}\}$.
Then for each
$g_{0}\in \overline{\Theta }$, there exists a sequence $\{\widetilde{f}%
_{n}\}_{n=1}^{\infty }\subset \widetilde{C}[a,b]$ such that
$\lim_{n\to \infty }\|\widetilde{f}_{n}\|=\infty $ and
$\lim_{n\to \infty}s_{\mathcal{P}}(\widetilde{f}_{n})=g_{0}$.
\end{theorem}


\paragraph{Proof} We define for each $n>2(b-a)^{-1}$ a
function $\varphi _{n}:[a,b]\to \mathbb{R}$ which
satisfies the following conditions:

\begin{itemize}
\item[$a)$]  $\varphi _{n}\in C^{1}[a,b]$

\item[$b)$]  $\varphi _{n}(a)=\varphi _{n}(b)$

\item[$c)$]  $\varphi _{n}(a+\frac{1}{2n})=\varphi _{n}(b-\frac{1}{2n})=1$

\item[$d)$]  $\varphi _{n}(t)=v_{0}$ for all $t\in [ a+\frac{1}{n},b-%
\frac{1}{n}]$

\item[$e)$]  $\int_{a}^{b}\varphi _{n}(t)dt=0$
\end{itemize}
Clearly, these functions exist. The rest of the proof is
analogous to that
of Theorem \ref{thm1}. We just change $w_{n}$ by $\varphi _{n}$ and
$s_{\mathcal{N}}(\widetilde{f})$ by $s_{\mathcal{P}}(\widetilde{f})$.
The only difference with the other proof is that now the graphs of the
functions $\varphi _{n}$ are not uniformly bounded, and this is
the reason because we need now to assume that $g$ is bounded.
\hfill$\diamondsuit$


\begin{corollary} \label{coro4}
Assume that $g=g(v)\in C(\mathbb{R})$ is bounded and $g_{0}\in \overline{g(%
\mathbb{R})}$. Then there exists a sequence $\{\widetilde{f}%
_{n}\}_{n=1}^{\infty }\subset \widetilde{C}[a,b]$ such that
$\lim_{n\to \infty }\|\widetilde{f}_{n}\|=\infty $ and
$\lim_{n\to \infty}s_{\mathcal{P}}(\widetilde{f}_{n})=g_{0}$.
\end{corollary}


\paragraph{Proof} In \cite[Corollary 4]{M} it is shown the
existence of solutions for $n=1$ whenever $g=g(v)$ is continuous.
Hence, it is enough to observe that if $g$ does not depend on the
variable $t$ then $\Theta =g(\mathbb{R})$. \hfill$\diamondsuit$
\medskip

We have proved that the limits $\lim_{\|\widetilde{f}\|\to
\infty }s_{\mathcal{N}}(\widetilde{f})$ and
$\lim_{\|\widetilde{f}\|\to \infty
}s_{\mathcal{P}}(\widetilde{f})$ never exist if
$\overline{\Theta}$ is not a single point. This makes natural to
ask if some weaker asymptotic results are possible. For example,
for which
functions $\widetilde{f}\in \widetilde{C}[a,b]$ do the radial limits $%
\lim_{k\to \infty }s_{\mathcal{N}}(k\widetilde{f})$ or $%
\lim_{k\to \infty }s_{\mathcal{P}}(k\widetilde{f})$ exist?
Now we prove a comparison result which will be helpful for the
computation of these limits.

\begin{lemma}[Comparison Principle] \label{lm5}
Let $k>0$ and $\widetilde{f}\in \widetilde{C}[a,b]$. If
$w_{\mathcal{N}}$ is a solution of the problem
\begin{equation}
\begin{gathered}
w'(t)+g(t,w(t))=s_{\mathcal{N}}(k\widetilde{f})+k\widetilde{f}(t),
\quad t\in (a,b) \\
w(a)=w(b)=0,
\end{gathered} \label{problema-k}
\end{equation}
where $w_{\mathcal{P}}$ is a solution of the problem
\begin{equation}
\begin{gathered}
w'(t)+g(t,w(t))=s_{\mathcal{P}}(k\widetilde{f})+k\widetilde{f}(t),
\quad t\in (a,b) \\
w(a)=w(b);\quad \int_{a}^{b}w(t)dt=0,
\end{gathered} \label{problema-p}
\end{equation}
$v_{\mathcal{N}}$ is the unique solution of
\begin{equation}
\begin{gathered}
v'(t)=\widetilde{f}(t), \quad t\in (a,b) \\
v(a)=v(b)=0
\end{gathered} ,  \label{trivial-k}
\end{equation}
and $v_{\mathcal{P}}$ is the unique solution of
\begin{equation}
\begin{gathered}
v'(t)=\widetilde{f}(t), \quad t\in (a,b) \\
v(a)=v(b); \quad \int_{a}^{b}v(s)ds=0,
\end{gathered}  \label{trivial-p}
\end{equation}
then $\|w_{\mathcal{N}}-kv_{\mathcal{N}}\|\leq (b-a)(M-m)$ and \ $\|w_{%
\mathcal{P}}-kv_{\mathcal{P}}\|\leq \frac{1}{2}(b-a)(M-m)$, where $%
m:=\inf_{(t,s)\in [ a,b]\times \mathbb{R}}g(t,s)$ and $%
M:=\sup_{(t,s)\in [ a,b]\times \mathbb{R}}g(t,s)$. \end{lemma}


\paragraph{Proof:}
Let $w_{\mathcal{N}}$ be a solution of (\ref{problema-k}) and let
$v_{\mathcal{N}}(t)=\int_{a}^{t}\widetilde{f}(s)ds$ be the
solution of (\ref{trivial-k}). Then
\begin{equation*}
w_{\mathcal{N}}(t)=k\int_{a}^{t}\widetilde{f}(s)ds+s_{\mathcal{N}}(k%
\widetilde{f})(t-a)-\int_{a}^{t}g(s,w_{\mathcal{N}}(s))ds
\end{equation*}
and
\begin{eqnarray*}
w_{\mathcal{N}}(t)-kv_{\mathcal{N}}(t) &=&s_{\mathcal{N}}(k\widetilde{f}%
)(t-a)-\int_{a}^{t}g(s,w_{\mathcal{N}}(s))ds \\
&=&\frac{t-a}{b-a}\int_{a}^{b}g(s,w_{\mathcal{N}}(s))ds-\int_{a}^{t}g(s,w_{%
\mathcal{N}}(s))ds.
\end{eqnarray*}
Hence
\begin{equation*}
|w_{\mathcal{N}}(t)-kv_{\mathcal{N}}(t)|\leq (b-a)(M-m),\quad
\text{ for all }t\in [ a,b]
\end{equation*}
since
\[
(t-a)m\leq
\frac{t-a}{b-a}\int_{a}^{b}g(s,w_{\mathcal{N}}(s))ds\leq (t-a)M
\]
and
\[
(t-a)m\leq \int_{a}^{t}g(s,w_{\mathcal{N}}(s))ds\leq (t-a)M .
\]
This completes the proof for the Neumann problem. For the periodic case
we must take into account that if $w_{\mathcal{P}}$ is a solution
of (\ref{problema-p}) and
\[
v_{\mathcal{P}}(t)=\int_a^t\widetilde{f}(s)ds-\frac{1}{b-a}\int_a^b
\int_a^t\widetilde{f}(s)ds\,dt
\]
is the solution of (\ref{trivial-p}) then
\begin{align*}
w_{\mathcal{P}}(t)
&=kv_{\mathcal{P}}(t)+s_{\mathcal{P}}(k\widetilde{f})(t-\frac{a+b}{2})+
\frac{1}{b-a}\int_a^b\int_a^tg(s,w_{\mathcal{P}}(s))ds\,dt\\
&\quad -\int_a^tg(s,w_{\mathcal{P}}(s)ds.
\end{align*}
After this, the proof is quite similar to that of the Neumann
problem. \hfill$\diamondsuit$ \smallskip

 In what follows we denote by $|A|$ the
Lebesgue measure of the set $A$.

\begin{theorem} \label{thm6}
Assume that  the limits $g(t,\pm \infty
):=\lim_{s\to \pm \infty }g(t,s)$ exist uniformly in $t\in [a,b]$.
Given $\widetilde{f}\in \widetilde{C}[a,b]$ and
$F(t)=\int_{a}^{t}\widetilde{f}(s)ds$, we have that\\
$(i)$  If $| \{t\in [ a,b]:F(t)=0\}| =0$ then
\begin{equation*}
\lim_{k\to \infty }s_{\mathcal{N}}(k\widetilde{f})=\frac{%
\int_{F^{-1}(0,+\infty )}g(t,+\infty )dt+\int_{F^{-1}(-\infty
,0)}g(t,-\infty )dt}{b-a}
\end{equation*}
$(ii)$  If $| \{t\in [ a,b]:F(t)=0\}| >0$ and
$g(t,s)=g(t,0)$ for all $(t,s)$ in
$[ a,b]\times [-(b-a)(M-m),(b-a)(M-m)]$, then
\begin{align*}
&\lim_{k\to \infty }s_{\mathcal{N}}(k\widetilde{f})\\
&= \frac{1}{b-a}\Big(\int_{F^{-1}(0,+\infty )}g(t,+\infty )dt
+\int_{F^{-1}(-\infty,0)}g(t,-\infty )dt
+\int_{F^{-1}(0)}g(t,0)dt\Big).
\end{align*}
\end{theorem}

\paragraph{Proof} It follows from Lemma \ref{lm5} that
\begin{equation}
kF(t)-(b-a)(M-m)\leq w_{\mathcal{N}}(t)\leq
kF(t)+(b-a)(M-m)\text{, for all }t\in [a,b]\text{;}
\label{desigualdad}
\end{equation}
where $F(t)=\int_{a}^{t}\widetilde{f}(s)ds$. We define the sets:
\begin{gather*}
A^{+} =\{t\in [a,b]:F(t)>0\}=F^{-1}(0,+\infty ) \\
A^{0} =\{t\in [a,b]:F(t)=0\}=F^{-1}(0) \\
A^{-} =\{t\in [a,b]:F(t)<0\}=F^{-1}(-\infty ,0)
\end{gather*}
Then
\begin{align*}
s_{\mathcal{N}}(k\widetilde{f})
&=\frac{1}{b-a}\int_{a}^{b}g(t,w_{\mathcal{N}}(t))dt\\
&=\frac{1}{b-a}\int_{A^{0}}g(t,w_{\mathcal{N}}(t))dt
+\frac{1}{b-a}\int_{A^{+}}g(t,w_{\mathcal{N}}(t))dt\\
&\quad+\frac{1}{b-a}\int_{A^{-}}g(t,w_{\mathcal{N}}(t))dt
\end{align*}
Now we will estimate each one of the integrals which appear in the
equality above.
First, using (\ref{desigualdad}) and the Lebesgue's dominated convergence
theorem we have
\begin{eqnarray*}
\lim_{k\to \infty }\frac{1}{b-a}\int_{A^{+}}g(t,w_{\mathcal{N}}(t))dt &=&\frac{1}{%
b-a}\int_{A^{+}}g(t,+\infty )dt \\
\lim_{k\to \infty }\frac{1}{b-a}\int_{A^{-}}g(t,w_{\mathcal{N}}(t))dt &=&\frac{1}{%
b-a}\int_{A^{-}}g(t,-\infty )dt.
\end{eqnarray*}
Second, under the assumption $(i)$ (i.e. $| A^{0}| =0$)
we have
\begin{equation*}
\frac{1}{b-a}\int_{A^{0}}g(t,w_{\mathcal{N}}(t))dt=0.
\end{equation*}
On the other hand, under the hypotheses of $(ii)$ (i.e.
$g(t,s)=g(t,0) $ for all $(t,s)\in [ a,b]\times [
-(b-a)(M-m),(b-a)(M-m)]$), we obtain from (\ref{desigualdad}) that
\begin{equation*}
-(b-a)(M-m)\leq w_{\mathcal{N}}(t)\leq (b-a)(M-m)
\end{equation*}
for all $t\in A^{0}$. Hence,
\begin{equation*}
\frac{1}{b-a}\int_{A^{0}}g(t,w_{\mathcal{N}}(t))dt=\frac{1}{b-a}\int_{A^{0}}g(t,0)dt.
\end{equation*}
Taking into account the two items above we complete the proof.
\hfill$\diamondsuit$

\begin{theorem} \label{thm7}
Assume that $g(t,s)$ is bounded and  that the limits
$g(t,\pm \infty ):=\lim_{s\to \pm \infty }g(t,s)$ exist
uniformly in $t\in [a,b]$. Given $\widetilde{f}\in
\widetilde{C}[a,b]$ and
\begin{equation*}
H(t)=\int_{a}^{t}\widetilde{f}(s)ds-\frac{1}{b-a}\int_{a}^{b}\Big(
\int_{a}^{t}\widetilde{f}(s)ds\Big) dt,
\end{equation*}
we have that:\\
$(i)$  If $| \{t\in [ a,b]:H(t)=0\}| =0$ then
\[
\lim_{k\to \infty }s_{\mathcal{P}}(k\widetilde{f})
=\frac1{b-a}\big(\int_{H^{-1}(0,+\infty )}g(t,+\infty )dt
+\int_{H^{-1}(-\infty,0)}g(t,-\infty )dt\Big)
\]
$(ii)$  If $| \{t\in [ a,b]:H(t)=0\}| >0$ and
$g(t,s)=g(t,0)$ for all $(t,s)$ in
$[ a,b]\times[ -\frac{b-a}{2}(M-m),\frac{b-a}{2}(M-m)]$ then
\begin{align*}
&\lim_{k\to \infty }s_{\mathcal{P}}(k\widetilde{f})\\
&=\frac1{b-a}\Big(\int_{H^{-1}(0,+\infty )}g(t,+\infty )dt
+\int_{H^{-1}(-\infty,0)}g(t,-\infty )dt+\int_{H^{-1}(0)}g(t,0)dt.\Big)
\end{align*}
\end{theorem}

The proof of this theorem is analogous to that of
Theorem \ref{thm6}, using the periodic case of the comparison principle.
The following result is a direct consequence of the theorems
above:

\begin{corollary} \label{coro8}
With the notation of Theorems \ref{thm6} and \ref{thm7}, if $g=g(s)$ does not depend
on the variable $t$ and there exists the limits $g(\pm \infty
):=\lim_{s\to \pm \infty }g(s)$ then
\begin{equation*}
\lim_{k\to \infty
}s_{\mathcal{N}}(k\widetilde{f})=\frac{g(+\infty )\left|
F^{-1}(0,+\infty )\right| +g(-\infty )\left| F^{-1}(-\infty
,0)\right| }{b-a}
\end{equation*}
whenever $\left| F^{-1}(0)\right| =0$ and
\begin{equation*}
\lim_{k\to \infty
}s_{\mathcal{P}}(k\widetilde{f})=\frac{g(+\infty )\left|
H^{-1}(0,+\infty )\right| +g(-\infty )\left| H^{-1}(-\infty
,0)\right| }{b-a}
\end{equation*}
whenever $\left| H^{-1}(0)\right| =0$.
\end{corollary}

The following proposition gives an estimation of the size of the
sets of functions with the property that the radial limits exists.

\begin{proposition} \label{prop9}
The sets
\begin{equation*}
\mathcal{F}=\{\widetilde{f}\in \widetilde{C}[a,b]:
F(t)=\int_{a}^{t} \widetilde{f}(s)ds\text{ satisfies }| F^{-1}(0)| =0\}
\end{equation*}
and
\begin{align*}
\mathcal{H}=&\Big\{\widetilde{f}\in \widetilde{C}[a,b]:
H(t)=\int_{a}^{t}\widetilde{f}(s)ds-\frac{1}{b-a}\int_{a}^{b}
\Big( \int_{a}^{t}\widetilde{f}(s)ds\Big) dt \\
& \text{ satisfies }| H^{-1}(0)| =0\Big\}
\end{align*}
are dense non-meager subsets of the Banach space
$\widetilde{C}[a,b]$.
\end{proposition}


\paragraph{Proof} Clearly, $\mathcal{F}$ is a dense subset of $\widetilde{C}[%
a,b]$, since $\widetilde{\Pi }=\Pi \cap \widetilde{C}[a,b]$ is dense in $%
\widetilde{C}[a,b]$, where $\Pi $ denotes the set of algebraic
polynomials,
and $\widetilde{\Pi }\setminus \{0\} \subset \mathcal{F}$. Now, we are going to prove that $%
\mathcal{F}$ has nonempty interior, which implies that
$\mathcal{F}$ is
non-meager. Of course, there is no loss of generality if we assume that
$[a,b]=[-1,1]$. Then $\widetilde{f}(t)=t$ belongs to $\mathcal{F}$. Let
$\widetilde{g}\in \widetilde{C}[-1,1]$ be such that
$\|\widetilde{f}-\widetilde{g}\|<\frac{1}{4}$ and let
$G(t)=\int_{-1}^{t}\widetilde{g}(s)ds$. Then
\begin{equation*}
\frac{t^{2}}{2}-\frac{t}{4}-\frac{3}{4}\leq G(t)\leq \frac{t^{2}}{2}+\frac{t%
}{4}-\frac{1}{4}\quad \text{ for all }t\in [ -1,1].
\end{equation*}
Thus, $G^{-1}(0)\subset \{-1\}\cup [ 1/2,1]$. If $\
\#G^{-1}(0)\geq 3$ then there are two points $x,y\in [
1/2,1]$ such that $G(x)=G(y)=0$ and Rolle's theorem implies that
$G'(t)=\widetilde{g}(t)$ vanishes
at some point $\xi \in [ 1/2,1]$, which is impossible since $\|%
\widetilde{f}-\widetilde{g}\|<\frac{1}{4}$. This implies that $%
\#G^{-1}(0)\leq 2$, so that $\widetilde{g}\in \mathcal{F}$ and
$\mathcal{F}$ has nonempty interior and proves the claim for the
set $\mathcal{F}$. Finally, the proof of the claim for the set
$\mathcal{H}$ follows from similar arguments. \hfill$\diamondsuit$

\paragraph{Remark} Note that when $g\in C^{1}(\mathbb{R})$,
it follows from \cite[Theorems 3.3 and 3.4]{C-D}  that
$s_{\mathcal{N}}(\cdot )$ and $s_{\mathcal{P}}(\cdot )$ are continuous
functionals so that
$\lim_{\|\widetilde{f}\|\to 0}s_{\mathcal{N}}(\widetilde{f})=s_{%
\mathcal{N}}(0)$ and $\lim_{\|\widetilde{f}\|\to 0}s_{\mathcal{P}}(%
\widetilde{f})=s_{\mathcal{P}}(0)$. Now, $s_{\mathcal{N}}(0)=s_{\mathcal{P}%
}(0)=g(0)$ since \cite[Theorem 3]{M}  guarantees that $w(t)=0$
is the unique solution of the systems
\begin{gather*}
w'(t)+g(w(t))=g(0), \quad t\in (a,b) \\
w(a)=w(b)=0 \quad \text{or}\quad w(a)=w(b);\quad \int_{a}^{b}w(t)dt=0
\end{gather*}

\paragraph{Acknowledgements}
The authors are very grateful to Prof. A. Ca\~{n}ada both for
proposing the problems considered in this note and for several
useful suggestions.

\begin{thebibliography}{0} \frenchspacing

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\end{thebibliography}

\noindent\textsc{Jose Mar\'{\i}a Almira} (e-mail: jmalmira@ujaen.es)\\
\textsc{Naira Del Toro} (e-mail: ndeltoro@ujaen.es)\\
Departamento de Matem\'{a}ticas\\
Universidad de Ja\'{e}n. E.U.P. Linares\\
C/ Alfonso X el Sabio, 28.\\
23700 Linares (Ja\'{e}n) Spain

\end{document}