\documentclass[twoside]{article} \usepackage{amsfonts} % used for R in Real numbers \pagestyle{myheadings} \markboth{ Asymptotic behaviour of the solvability set } {A. Ca\~nada \& A. J. Ure\~{n}a} \begin{document} \setcounter{page}{55} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent USA-Chile Workshop on Nonlinear Analysis, \newline Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 55--64\newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Asymptotic behaviour of the solvability set for pendulum-type equations with linear damping and homogeneous Dirichlet conditions % \thanks{ {\em Mathematics Subject Classifications:} 34B15, 70K30. \hfil\break\indent {\em Key words:} Pendulum-type equations, linear damping, Dirichlet boundary conditions, \hfil\break\indent solvability set, asymptotic results, Riemann-Lebesgue lemma, Baire category. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Published January 8, 2001. \hfil\break\indent Supported by DGES, by grant PB98-1343 from the Spanish Ministry of Education and \hfil\break\indent Culture, and by grant FQM116 from Junta de Andaluc\'{\i}a.} } \date{} \author{ A. Ca\~nada \& A. J. Ure\~{n}a } % A. Canada \& A. J. Urena \maketitle \begin{abstract} We show some results on the asymptotic behavior of the solvability set for a nonlinear resonance boundary-value problem, with linear damping, periodic nonlinearity and homogeneous Dirichlet boundary conditions. Our treatment of the problem depends on a multi-dimensional generalization of the Riemann-Lebesgue lemma. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12 \section{Introduction} Solvability of the nonlinear boundary-value problem \begin{equation}\label{eqno11} \begin{array}{c} -u''(x) - \alpha u'(x) - \lambda_1(\alpha)u(x) + g(u(x)) = h(x), \quad x \in [0,\pi], \\[2pt] u(0) = u(\pi) = 0\,, \end{array} \end{equation} has been studied by several authors under the following set of hypotheses. \begin{description} \item{\bf [H]} $\alpha$ is a given real number, $\lambda_1(\alpha) = 1 + \alpha^2/4$ is the first eigenvalue of the eigenvalue problem \begin{equation}\label{eqno12} \begin{array}{c} -u''(x) - \alpha u'(x) = \lambda u(x), \quad x \in [0,\pi] \\[2pt] u(0) = u(\pi) = 0\,, \end{array} \end{equation} $g: {\mathbb R} \to {\mathbb R}$ is a continuous and $T$-periodic function with zero mean value, and $h \in L^{1}[0,\pi]$ \,. \end{description} \noindent The case $\alpha = 0$ can be found in \cite{car1,danc, scsc, ward}, while the case $\alpha \neq 0$ has been recently treated in \cite{caur1}. These type of problems, with periodic nonlinearity, are important in applications and (\ref{eqno11}) models, for example, the motion of a pendulum clock (\cite{jord, maw1}). If $g$ is not identically zero and $\psi (x) = \exp(\alpha x/2)\sin (x)$ is the principal positive eigenfunction of the adjoint problem to (\ref{eqno12}) for $\lambda = \lambda_1(\alpha)$, it was proven in (\cite{caur1}) that for a given $ \tilde{h} \in L^1[0,\pi]$, with $\int_{0}^{\pi} \tilde{h}(x)\psi (x) \,dx = 0$, there exist real numbers $a_1(\tilde{h}) < 0 < a_2(\tilde{h})$, such that (\ref{eqno11}), with $h$ given by $h(x) = a \psi (x) + \tilde{h} (x)$, $(a\in{\mathbb R})$, has solution if, and only if, $a \in [a_1(\tilde{h}),a_2(\tilde{h})]$. However, very little is known on the behavior of the functionals $a_1$ and $a_2$. In this paper we deal with their asymptotic behavior. More precisely, we shall show that if $$ \tilde{L}^1[0,\pi] = \Big\{ h \in L^1[0,\pi] : \int_{0}^{\pi} h(x)\psi (x) \,dx = 0 \Big\}, $$ then there exist a subset $F \subset\tilde{L}^1[0,\pi]$, (which will be explicitly described) of first category in $\tilde{L}^1[0,\pi]$ in the sense of Baire, such that for each $\tilde{h} \in \tilde{L}^1[0,\pi] \setminus F$, \begin{equation}\label{eqno14} \lim _{\vert \lambda \vert \to \infty} a_1(\lambda \tilde{h} ) = \lim _{\vert \lambda \vert \to \infty} a_2(\lambda \tilde{h} ) = 0\,. \end{equation} As a trivial consequence, the set of functions $\tilde{h} \in \tilde{L}^1[0,\pi]$ for which (\ref{eqno14}) is true, is a dense and second category subset of $\tilde{L}^1[0,\pi]$. In the final remarks we briefly comment why this result cannot be strengthened very much, since, under hypotheses [H], it may happens that (\ref{eqno14}) does not occur also for a dense subset of $\tilde{L}^1[0,\pi]$ (see \cite{caur2}). Let us point out that related results for the case of periodic boundary conditions and $\alpha = 0$ can be found in \cite{kaor}. However, to the best of our knowledge, properties like (\ref{eqno14}) for the problem (\ref{eqno11}) and periodic nonlinearity $g$, have not been previously treated in the literature, even for the case $\alpha = 0$. In the proofs we use the Liapunov-Schmidt reduction, The Baire's category theorem, some notions on measure theory and the multi-dimensional version of the Riemann-Lebesgue lemma developed in Lemma 3.1 (see \cite{danc,kaor,zygm} for the classical one-dimensional version). Through this paper, $\langle\cdot,\cdot\rangle$ will stand for the Euclidean inner product in ${\mathbb R}^N$, while for any $x\in{\mathbb R}^N$, $\|x\|:=\sqrt{\langle x,x\rangle}$ will denote its associated norm and $x_1,\dots ,x_N$ its components. We will write as $\|\cdot\|_1$ and $\|\cdot\|_\infty$ the usual norms in $L^1[0,\pi]$ and $L_\infty[0,\pi]$ respectively. A function $h\in L^1[0,\pi]$ will be called a {\em step function} if there exists a partition $0 = x_{0} < x_1<\dots 0$ independent of $\lambda \in {\mathbb R}$, such that \begin{equation}\label{eqno34} \begin{array}{c} \vert K(I-Q)N_{0}(c_\lambda \varphi + u_\lambda )(x) \vert \leq M, \ \forall \ x \in [0,\pi], \\[2pt] \vert (K(I-Q)N_{0}(c_\lambda \varphi + u_\lambda ))'(x) \vert \leq M, \ \forall \ x \in [0,\pi] \end{array} \end{equation} Previous discussion motivates the next multidimensional generalization of the Riemann-Lebesgue lemma. \begin{lemma} \label{l31} Let $g:{\mathbb R} \to {\mathbb R}$ be a continuous and $T$-periodic function with zero mean value and let $u_1,\dots ,u_N \in C^{1}[0,\pi]$ be given functions satisfying the following property: \begin{description} \item{\bf [P]} If $\rho_1,\dots ,\rho_N$ are real numbers such that $$ \mathop{\rm meas } \Big\{ x \in [0,\pi]: \sum_{i=1}^{N} \rho_i u_i '(x) = 0 \Big\} > 0\,, $$ then $\rho_1 = \dots = \rho_N = 0$. \end{description} Let $B \subset C^{1}[0,\pi]$ be such that the set $\{ b', \ b \in B \} $ is uniformly bounded in $C[0,\pi].$ Then, for any given function $r \in L^1[0,\pi]$, we have \begin{equation}\label{eqno35} \lim_{\Vert \rho \Vert \to \infty} \int_{0}^{\pi} g \left ( \sum_{i=1}^{N} \rho_i u_i (x) + b(x) \right ) r(x) \,dx = 0 \end{equation} uniformly with respect to $b \in B$. \end{lemma} \paragraph{Proof.} Let $r \in L^1[0,\pi] $ be a given function and let $\{ \rho^{n}, \ n \in {\mathbb N} \}\subset {\mathbb R}^{N}$ and $\{ b^{n}, \ n \in {\mathbb N} \} \subset B$ be given sequences with $\Vert \rho^{n} \Vert \to \infty$. If we define $\mu^{n} = \rho^{n}/\Vert \rho^{n} \Vert$, we have, at least for a subsequence, that $\mu^{n} \to \mu $ for some $\mu\in{\mathbb R}^N$ with $\mu_1^{2}+\dots +\mu_{N}^{2} = 1$. If $u = (u_1,\dots ,u_N )$, then by hypothesis, $\mathop{\rm meas}(Z) = 0$, where $Z = \{ x \in [0,\pi]: \langle \mu,u'(x)\rangle = 0 \}$. This implies that the linear span of the set \begin{equation}\label{s11} S = \{ \langle \mu,u'\rangle \chi_{I}: \ I \mbox{ is any compact subinterval of } [0,\pi], I \cap Z = \emptyset \} \end{equation} is a dense set in $L^1[0,\pi]$. To see this, let us define \begin{equation}\label{s12} S_1 = \{ \chi_{I}: \ I \mbox{ is any compact subinterval of } [0,\pi], I \cap Z = \emptyset \} \end{equation} Then, for any open subset $A \subset [0,\pi]$ (in particular, for any open subinterval of $[0,\pi]$), $\mathop{\rm meas}(A\setminus Z) = \mathop{\rm meas}(A)$. Since $A\setminus Z$ is also open, there exists an at most countable collection $ \{ I_{i}, \ i \in {\mathbb N} \} $ of pairwise disjoint open intervals such that $A\setminus Z = \cup_{i\in {\mathbb N}} \ I_{i}$ and $\mathop{\rm meas}(A\setminus Z) = \sum_{i \in {\mathbb N}} \mathop{\rm meas}(I_{i})$. Consequently, the linear span of the set $S_1$ is a dense set in the set of step functions and therefore in $L^1[0,\pi]$. Now, let $\chi_{I}$ be a given element of $S_1$. Write $ w = \langle \mu,u'\rangle $ and $m = \inf_{I} \ \vert w \vert $ ($m>0$). Finally, fix $\epsilon>0$. Choose a partition of $I = [a,b]$, $a = a_{0}< a_1< \dots < a_{m-1} < a_{m} = b$ such that if $x,y \in J_{i} =[a_{i-1},a_{i}]$, $1 \leq i \leq m$, then $\vert w(x) - w(y) \vert \leq \epsilon$. Then, for any $x \in I$, there is some $i$, $1 \leq i \leq m$, such that $x \in J_{i}$ and $$ \left \vert \chi_{I}(x) - \sum_{i=1}^{m} \frac{w\chi_{J_{i}}(x)}{w(a_{i})} \right | = \left | \frac{w(a_{i}) - w(x)}{w(a_{i})} \right | \leq \epsilon/m, $$ so that $$\Big\|\chi_{I} - \sum_{i=1}^{m} \frac{w\chi_{J_{i}}}{w(a_{i})} \Big\|_1\leq \epsilon\pi/m\,. $$ Consequently, we deduce that the linear span of $S$ is dense in $S_1$ and therefore in $L^1[0,\pi]$. On the other hand, if $l^{\infty}$ denotes the space of bounded sequences of real numbers with the usual norm, the linear operator $T: L^1[0,\pi] \to l^{\infty}$, $ s \to \{ (Ts)^{n}, \ n \in {\mathbb N} \}$, defined by $$ (Ts)^{n} = \int_{0}^{\pi} g(\langle \rho^{n},u(x)\rangle + b^{n}(x))s(x)\ dx, \ \forall \ s \in L^1[0,\pi], \ \forall \ n \in {\mathbb N}, $$ is trivially continuous. Recall that our purpose is to prove that $T(L^1[0,\pi]) \subset l_{0}$, the closed subspace of $l^{\infty }$ of all sequences which converge to zero. Since $T$ is continuous and $l_{0}$ is closed, to prove the lemma it is sufficient to demonstrate that $T(S) \subset l_{0}$, i.e., \begin{equation}\label{eqno36} \lim_{n \to \infty} \ \int_{I} \ g(\langle \rho^{n},u(x)\rangle + b^{n}(x))(\langle \mu,u'(x)\rangle ) \,dx = 0, \end{equation} for any compact subinterval $I$ of $[0,\pi]$ such that $I\bigcap Z = \emptyset$. But, if $v^{n}, v: I \to {\mathbb R}$ are defined as $$\begin{array}{c} v^{n}(x) = \langle \mu^{n},u(x)\rangle + b^{n}(x)/\Vert \rho^{n} \Vert, \\[2pt] v(x) = \langle \mu,u(x)\rangle , \ \forall \ x \in [0,\pi], \end{array} $$ we trivially have \begin{equation}\label{eqno37} \lim _{n \to \infty} \int_{I} g(\Vert \rho^{n} \Vert v^{n}(x))(v'(x) - (v^{n})'(x)) \,dx = 0 \end{equation} and \begin{eqnarray}\label{eqno38} \lefteqn{\lim _{n \to \infty} \int_{I} g(\Vert \rho^{n} \Vert v^{n}(x))(v^{n})'(x)) \,dx } \\ &=& \lim_{n \to \infty} \frac{G(\Vert \rho^{n} \Vert v^{n}(\max I))- G(\Vert \rho^{n} \Vert v^{n}(\min I))}{\Vert \rho^{n} \Vert } = 0 \nonumber \end{eqnarray} where $G$ is any primitive function of function $g$. Now, (\ref{eqno37}) and (\ref{eqno38}) imply (\ref{eqno36}). \paragraph{Remark.} It is clear that both the conclusion and the proof of the previous lemma are still true under more general hypotheses on the function $g$. It is sufficient that $g$ be a continuous and bounded function with bounded primitive $G$. Next, we apply the previous lemma to the specific problem of the asymptotic behavior of the functional $a$ whose expression was given in (\ref{eqno33}). \begin{corollary}\label{c31} Let $\tilde{h} \in \tilde{L}^1[0,\pi]$ be a given function and suppose that the functions $K\tilde{h}$ and $\varphi$ satisfy the following property \begin{description} \item{\bf [P1]} If $\rho_1,\rho_2$ are real numbers such that $$ \mathop{\rm meas } \{ x \in [0,\pi]: \rho_1 (K\tilde{h})'(x) + \rho_2 \varphi'(x) = 0 \} > 0, $$ then $\rho_1 = \rho_2 = 0$. \end{description} Let $B \subset \tilde{L}^1[0,\pi]$ be any bounded subset. Then \begin{equation}\label{eqno39} \lim_{\vert \lambda \vert \to \infty} \ a(\lambda \tilde{h} + b) = 0, \end{equation} uniformly with respect to $b \in B$. \end{corollary} \paragraph{Proof.} For each $\lambda \in {\mathbb R}$ and each $b \in B$, there is $(c_{\lambda,b},\tilde{u}_{\lambda,b}) \in \Sigma_{\lambda,b}$ such that \begin{eqnarray} a(\lambda \tilde{h} + b)\int_{0}^{\pi} (\psi (x))^{2} \,dx &=& \int_{0}^{\pi} g(c_{\lambda,b}\varphi(x) + \lambda K\tilde{h} (x)+ Kb (x) \label{eqno310} \\ &&+ K(I-Q)N_{0}(c_{\lambda,b}\varphi + \tilde{u}_{\lambda,b})(x)) \psi (x) \,dx \nonumber \end{eqnarray} where $\Sigma_{\lambda,b}$ is the corresponding solution set of the auxiliary equation for $\lambda \tilde{h} + b$. Since the set $$ \{ Kb+ K(I-Q)N_{0}(c_{\lambda,b}\varphi + \tilde{u}_{\lambda,b}), \ \lambda \in {\mathbb R}, \ b \in B \} $$ is bounded in $C^{1}[0,\pi]$ (see (\ref{eqno34})), the conclusion follows from the previous lemma. The following equivalent version of previous corollary will be very useful for our purposes. \begin{corollary} \label{c32} Let $\tilde{h} \in \tilde{L}^1[0,\pi]$ be a given function and suppose that, for every $\rho \in {\mathbb R}$, $$ \mathop{\rm meas} \{ x \in [0,\pi]: \ (K\tilde{h})'(x) = \rho \varphi'(x) \} = 0. $$ Let $B \subset \tilde{L}^1[0,\pi]$ be any bounded subset. Then \begin{equation}\label{eqno39bis} \lim_{\vert \lambda \vert \to \infty} \ a(\lambda \tilde{h} + b) = 0, \end{equation} uniformly with respect to $b \in B.$ \end{corollary} Now, we state and prove our main result. \begin{theorem} \label{t31} There exists a subset $F \subset \tilde{L}^1[0,\pi]$, of first category in $\tilde{L}^1[0,\pi]$, such that for any given $\tilde{h} \in \tilde{L}^1[0,\pi] \setminus F$, and each given bounded subset $B \subset \tilde{L}^1[0,\pi]$, one obtains \begin{equation}\label{eqno311} \lim_{\vert \lambda \vert \to \infty} \ a(\lambda \tilde{h} + b) = 0 \end{equation} uniformly with respect to $b \in B$. \end{theorem} \paragraph{Proof.} Let $$ F = \left \{ \tilde{h} \in \tilde{L}^1[0,\pi]: \ \exists \ \rho \in {\mathbb R} \mbox{ s.t.\ meas } \{ x \in [0,\pi]: (K\tilde{h})'(x) = \rho \varphi'(x) \} > 0 \right \} $$ Then $ F = \cup_{n\in {\mathbb N}} \ F_{n}$, where $$ F_{n} = \left \{ \tilde{h} \in \tilde{L}^1[0,\pi]: \ \exists \ \rho \in {\mathbb R} \mbox{ s.t.\ meas} \{ x \in [0,\pi]: (K\tilde{h})'(x) = \rho\varphi'(x) \} \geq 1/n \right \} $$ Let us prove that each subset $F_{n}$ is closed and has an empty interior. To see this, let us fix $F_{n}$. Then, since $K:\mathop{\rm ker} Q \to\mathop{\rm ker}P$ is a topological isomorphism, $F_{n}$ is a closed subset of $\mathop{\rm ker} Q$ if and only if $K(F_n) \equiv G_{n}$ is a closed subset of $\mathop{\rm ker} P$. Now, it is clear that $G_{n}$ is the set of functions $$\{ u \in\mathop{\rm ker} P: \exists \rho \in {\mathbb R} \mbox{ s.t.\ meas} \{ x \in [0,\pi]: u'(x) = \rho \varphi'(x) \} \geq 1/n \} $$ Let $\{u_{m}, \ m \in {\mathbb N} \} \subset G_{n}$ be a sequence such that $\{u_{m} \} \to u$ in $\mathop{\rm ker} P$. Then, for any $m \in {\mathbb N}$, we can find $\rho_{m}\in {\mathbb R}$ such that $$ \mathop{\rm meas} \{ x \in [0,\pi]: u'_{m}(x) = \rho_{m}\varphi'(x) \} \geq 1/n $$ Since $$ \mathop{\rm meas} \{ x \in [0,\pi]: \varphi'(x) = 0 \} = 0, $$ the sequence $\{ \rho_{m} \}$ must be bounded and, after possibly passing to a subsequence, we can suppose, without loss of generality, that $\{ \rho_{m} \} \to \rho$. Moreover, if we define $$ M_{m} = \{ x \in [0,\pi]: u'_{m}(x) = \rho_{m}\varphi'(x) \} $$ then $\mathop{\rm meas} M_{m} \geq 1/n, \ \forall \ m \in {\mathbb N}$ and $\mathop{\rm meas} \left (\bigcap_{m=1}^{\infty} \left [ \bigcup_{s=m}^{\infty} M_{s} \right ] \right ) \geq 1/n$. Finally, let us observe that if $x \in \bigcap_{m=1}^{\infty} \left [ \bigcup_{s=m}^{\infty} M_{s} \right ]$, then $u'(x) = \rho \varphi'(x)$, so that $\mathop{\rm meas} \{x \in [0,\pi]: u'(x) = \rho \varphi'(x) \} \geq 1/n$ and consequently $u \in G_{n}$. Next, we prove that $F$ (and therefore each $F_n$) has an empty interior. To see this, let us define the function $\varphi_1$ as the primitive of $\varphi$ with zero mean value and $\varphi_2$ as the primitive of $\varphi_1$ satisfying $\varphi_2 (0) = \varphi_2 (\pi) = 0$. Then, $\varphi_2 \in W_0^{2,1}[0,\pi], \ \varphi_2 '' = \varphi$ and for any $u \in W_0^{2,1}[0,\pi]$, we have $$ \int_{0}^{\pi} u\varphi = - \int_{0}^{\pi} u'\varphi_1 = \int_{0}^{\pi} u'' \varphi_2. $$ As a consequence, the mapping $\Phi:\mathop{\rm ker} P \to \tilde{L}^1_{\varphi_2}[0,\pi] $, $u \to u''$ is a topological isomorphism, where $$ \tilde{L}^1_{\varphi_2}[0,\pi] = \{ h \in L^1[0,\pi]: \int_{0}^{\pi} h(x) \varphi_2 (x) \,dx = 0 \} $$ Therefore, $F$ has an empty interior in $\tilde{L}^1[0,\pi]$ provided $\Phi(K(F))$ has an empty interior in $\tilde{L}^1_{\varphi_2}[0,\pi]$. This last result is an easy consequence of the following lemma. \begin{lemma} Let us denote by $A$ the subset of $L^1[0,\pi]$ given by all the step functions and by $B$ the subset of $L^1[0,\pi]$ given by all the non-vanishing step functions. Then, \begin{enumerate} \item The set $A \cap \tilde{L}^1_{\varphi_2}[0,\pi] $ is dense in $\tilde{L}^1_{\varphi_2}[0,\pi]$. \item The set $B \cap \tilde{L}^1_{\varphi_2}[0,\pi] $ is dense in $\tilde{L}^1_{\varphi_2}[0,\pi]$. \item $ B \cap \Phi(K(F)) = \emptyset$ \end{enumerate} \end{lemma} \paragraph{Proof.} \begin{enumerate} \item Let us choose any $h\in \tilde{L}^1_{\varphi_2}[0,\pi]$ and $\epsilon > 0$. Then, there exists $s \in A$ such that $\Vert h-s \Vert_1 < \min \ \{ \epsilon/2\pi, \ \frac{\Vert \varphi_2 \Vert_1}{\Vert \varphi_2 \Vert_{\infty}} \}.$ Now, the function $\tilde{s} = s + \frac{\int_{0}^{\pi} s \varphi_2 }{\Vert \varphi_2 \Vert_1}$ is again a step function which belongs to $\tilde{L}^1_{\varphi_2}[0,\pi]$ and such that $\Vert h - \tilde{s} \Vert_1 < \epsilon$. \item Let us demonstrate that $B \cap \tilde{L}^1_{\varphi_2}[0,\pi]$ is dense in $A \cap \tilde{L}^1_{\varphi_2}[0,\pi]$. To see this, let us take $u \in A \cap \tilde{L}^1_{\varphi_2}[0,\pi]$. If $a,b \in {\mathbb R}$, define the function $u_{a,b} = u + a\chi_{[0,\pi/2]} + b \chi_{[\pi/2,\pi]}$. The condition for $u_{a,b}$ to belong to $\tilde{L}^1_{\varphi_2}[0,\pi]$ is $$ a \int_{0}^{\pi/2} \varphi_2 + b \int_{\pi/2}^{\pi} \varphi_2 = 0 $$ Since $\int_{0}^{\pi/2} \varphi_2 < 0$ and $\int_{\pi/2}^{\pi} \varphi_2 < 0,$ (think that, by the maximum principle, $\varphi_2 < 0$ in $(0,\pi)$), it is clear that we may choose $a$ and $b$ both different from zero but as small as we want in absolute value such that $u_{a,b} \in B \cap \tilde{L}^1_{\varphi_2}[0,\pi]$. \item If $s \in B \cap \Phi(K(F))$, then there is $\tilde{h} \in F$ such that $K(\tilde{h}) = u, \ \Phi (u) = u'' = s$. Since $\tilde{h} \in F$, there exists $\rho \in {\mathbb R}$ such that meas $ \{ x \in [0,\pi]: u'(x) = \rho \varphi'(x) \} \ > 0$. Choose some nontrivial compact interval $I \subset [0,\pi]$ satisfying $s |_{I} \equiv c \neq 0$ and such that meas $ \{ x \in I: u'(x) = \rho \varphi'(x) \} \ > 0$. This implies that meas $ \{ x \in I: c = u''(x) = \rho \varphi''(x) \} \ > 0$, which is a contradiction with the form of the function $\varphi$. \end{enumerate} \paragraph{Final Remark.} Under the hypotheses [H], it is possible to show that, in many cases, the set of functions $\tilde{h} \in \tilde{L}^1[0,\pi]$ for which $\lim_{\vert \lambda \vert \to \infty} \ a(\lambda \tilde{h})$ is not zero, is also dense in $\tilde{L}^1[0,\pi]$. For example, this is true for the oscillating function $\delta g$ in the place of $g$ provided that $|\delta|$ is small enough. In this case, it may be proved that the previous limit is not zero if the function $ u = K(\tilde{h})$ belongs to the set of functions in $\mathop{\rm ker} P$ for which there exists a partition $ 0 = x_{0}< x_1 < \dots < x_{p-1} < x_{p} = \pi$ and $1 \leq i_{0} \leq p$ and constants $\mu \neq 0, \ c \neq 0$, such that \begin{description} \item{i)} $u''_{[x_{i-1},x_{i}]}$ is a constant function, for any $1 \leq i\leq p$, $i \neq i_{0}$. \item{ii)} $u(x) = \mu \varphi(x) + c, \ \forall \ x \in [x_{i_{0}-1},x_{i_{0}}]$. \end{description} After this, it may be proved that this set is dense in $\mathop{\rm ker} P$. The detailed proof may be found in \cite{caur2}. \begin{thebibliography}{99} \bibitem{car1} {A. Ca\~{n}ada and F. Roca,} {Existence and multiplicity of solutions of some conservative pendulum-type equations with homogeneous Dirichlet conditions.} {Diff. Int. Eqns., 10, (1997), 1113-1122.} \bibitem{caur1} {A. Ca\~{n}ada and A.J. 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An\'alisis Matem\'atico, Univ. de Granada,\\ 18071-Granada, Spain \\ e-mail: ajurena@ugr.es \end{document}