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\begin{document}
%\doi{10.1080/1023619YYxxxxxxxx}
% \issn{1563-5120}
%\issnp{1023-6198} \jvol{00} \jnum{00} \jyear{2008} \jmonth{January}

\markboth{A. Krist\'aly, M. Mih\u ailescu and V. R\u
adulescu}{Small and large solutions for discrete BVPs}

%\articletype{GUIDE}

\title{{\itshape Discrete boundary value problems involving oscillatory nonlinearities: small and large solutions}}

\author{Alexandru  Krist\'aly$^{\rm a}$,  Mihai Mih\u ailescu
$^{\rm {b,c}}$ and Vicen\c tiu R\u adulescu $^{\rm
{c,d,}}$$^{\ast}$\thanks{$^\ast$Corresponding author. Email:
vicentiu.radulescu@math.cnrs.fr}\\ \vspace{6pt}  $^{\rm
a}${\em{Babe\c s-Bolyai University, Department of Economics,
400591 Cluj-Napoca, Romania}};\\ $^{\rm b}${\em{Department of
Mathematics, Central European University, 1051 Budapest,
Hungary}}; \\ $^{\rm c}${\em{Department of Mathematics, University
of Craiova, 200585 Craiova, Romania}}; \\ $^{\rm d}${\em{Institute
of Mathematics ``Simion Stoilow" of the Romanian Academy,\\ \small
P.O. Box 1-764, 014700 Bucharest, Romania}}
\vspace{6pt}\received{v3.5 released October 2008}
%\thanks{A. Krist\'aly. Email: alexandrukristaly@yahoo.com}
\\\vspace{6pt}\received{Submitted: February 2009} }



\maketitle

\begin{abstract}
We consider the discrete boundary value problem (P):
$-\Delta(\Delta u(k-1))= f(u(k)),\ k\in[1,T],$ $ u(0)=u(T+1)=0$,
where the nonlinear term $f:[0,\infty)\to \mathbb R$ has an
oscillatory behaviour near the origin or at infinity. By a direct
variational method we show that (P) has a sequence of
non-negative, distinct solutions which converges to 0 (resp.
$+\infty$) in the sup-norm whenever $f$ oscillates at the origin
(resp. at infinity).
\bigskip

\begin{keywords}difference equations; oscillatory nonlinearities; small solutions; large solutions
\end{keywords}
\begin{classcode} 46E39, 34K10, 35B38 \end{classcode}\bigskip
\end{abstract}


\section{Introduction and main results}
%Discrete and continuous features coexist in nature depending on
%the scales of observation.
In many cases a problem in a continuous framework can be handled
by using a suitable method from discrete mathematics and
conversely; a beautiful description of such phenomena can be found
in Lov\'asz \cite{LL}. The modeling/simulation of certain
nonlinear problems from economics, biological neural networks,
optimal control and others enforced in a natural manner the rapid
development of the theory of difference equations. The reader may
consult the comprehensive monographs of Agarwal \cite{A},
Kelley-Peterson \cite{KP}, Lakshmikantham-Trigiante \cite{LT}.

Within the theory of difference equations, a large class of
problems is the nonlinear discrete boundary value problems. To be
more precise, we consider the problem $$ \left\{\begin{array}{lll}
-\Delta(\Delta u(k-1))=f(u(k)), \quad k\in[1,T],\\ u(0)=u(T+1)=0,
 \end{array}\right.\eqno{\rm (P)}$$
 where $T\geq 2$ is an integer, $[1,T]$ is the discrete interval
$\{1,...,T\}$, $\Delta u(k)=u(k+1)-u(k)$ is the forward difference
operator, and $f$ is a continuous nonlinearity. In order to
establish existence/multiplicity of solutions for (P) under
specific restrictions on $f$ (sublinear or superlinear at
infinity), the authors exploited various abstract methods as fixed
point theorems, sub- and super-solution arguments, Brouwer degree
and critical point theory. We refer the reader to the recent
papers of Agarwal-Perera-O'Regan \cite{APR,APR1}, Bereanu-Mawhin
\cite{Ber-Maw-1, Ber-Maw-2}, Bereanu-Thompson \cite{BT},
Bonanno-Candito \cite{BC}, Cabada-Iannizzotto-Tersian \cite{cit},
Cai-Yu \cite{CY}, Mih\u ailescu-R\u adulescu-Tersian \cite{mrt},
Yu-Guo \cite{YG}, Tang-Luo-Li-Ma \cite{4kinai}, Zhang-Liu
\cite{ZL}, and references therein.

The main purpose of the present paper is to trait problem (P) when
the nonlinear term $f:[0,\infty) \to \mathbf R$ has a suitable
{\it oscillatory behaviour}. A direct variational argument
provides two results (see Theorems \ref{result-zero-osc} and
\ref{result-infinit-osc}), guaranteeing sequences of non-negative
solutions with further asymptotic properties whenever $f$
oscillates near the origin or at infinity. Before to state our
results, we mention that solutions of ${\rm (P)}$ are going to be
sought in the function space $$X=\{u:[0,T+1]\to \mathbb R;\
u(0)=u(T+1)=0\}.$$
 Clearly, $X$ is a
$T$-dimensional Hilbert space (see \cite{APR}) with the inner
product $$\langle u,v\rangle =\sum_{k=1}^{T+1}\Delta u(k-1)\Delta
v(k-1),\;\;\;\forall\;u,v\in X\,.$$ The associated norm is defined
by $$\|u\|=\left(\sum_{k=1}^{T+1}|\Delta
u(k-1)|^2\right)^{1/2}\,.$$ The space $X$ being
finite-dimensional, the sup-norm $\|\cdot\|_\infty$ is equivalent
to $\|\cdot\|$; here, we denote $\|u\|_\infty=\max_{k\in
[1,T]}|u(k)|$, $u\in X.$

In the sequel, we state our results. Let $F(s)=\int_0^sf(t)dt,$
$s\in [0,\infty).$

Our first result concerns the case when $f$ has a certain type of
oscillation near the origin. To be more precise, we assume
\begin{enumerate}
\item[$(H^0)$] \ \ \
  $\liminf_{s\to 0^+}\frac{f(s)}{s}< 0$; $\limsup_{s\to
0^+}\frac{F(s)}{s^2}>\frac{1}{T}$.
\end{enumerate}



\begin{theorem}
%{\rm (Oscillation at the origin)}
\label{result-zero-osc} Let $f\in C^0([0,\infty); \mathbf R)$
verifying $(H^0)$. Then there exists a sequence
$\{u_n^0\}_{n}\subset X$ of non-negative, distinct solutions of
${\rm (P)}$ such that
\begin{equation}\label{elsobecslesnullaban}
  \lim_{n\to \infty}\|u_n^0\|_{\infty}=\lim_{n\to \infty}\|u_n^0\|=0.
\end{equation}
\end{theorem}


\noindent A perfect counterpart of Theorem \ref{result-zero-osc}
can be stated when the nonlinear term oscillates at infinity.
Instead of $(H^0)$, we assume
\begin{enumerate}
\item[$(H^\infty)$] \ \ \
  $\liminf_{s\to \infty}\frac{f(s)}{s}< 0$; $\limsup_{s\to
\infty}\frac{F(s)}{s^2}>\frac{1}{T}$.
\end{enumerate}



\begin{theorem}  \label{result-infinit-osc} %{\rm (Oscillation at infinity)}
   Let $f\in C^0([0,\infty); \mathbf R)$
verifying $(H^\infty)$ and $f(0)=0$. Then there exists a sequence
$\{u^{\infty}_n\}_{n}\subset X$ of non-negative, distinct
solutions of ${\rm (P)}$ such that
\begin{equation}\label{vegtelen-asszimpt}
 % \lim_{k\to
%\infty}\|u_k^\infty\|_a=
\lim_{n\to \infty}\| u_n^\infty\|_{\infty}=\lim_{n\to \infty}\|
u_n^\infty\|=\infty.
\end{equation}
\end{theorem}

\begin{example} (a) Let
$\alpha,\beta,\gamma\in \mathbf R$ such that
$0<\alpha<1<\alpha+\beta,$   and $\gamma\in (0,1).$ Then, the
function $f:[0,\infty)\to \mathbf R$ defined by $f(0)=0$ and
$f(s)=s^{\alpha}(\gamma+\sin s^{-\beta})$, $s>0$, verifies
hypothesis $(H^0)$.

(b) Let $\alpha,\beta,\gamma\in \mathbf R$ such that $1<\alpha,$
$|\alpha-\beta|<1$, and $\gamma\in (0,1).$ Then, the function
$f:[0,\infty)\to \mathbf R$ defined by
$f(s)=s^{\alpha}(\gamma+\sin s^{\beta})$ verifies $(H^\infty)$.
\end{example}

 The paper is divided as follows. In the next
section we consider a related difference equation to ${\rm (P)}$;
the existence of a non-negative solution is proved under some
generic assumptions. This result is used in Sections 3 and 4,
where Theorems \ref{result-zero-osc} and \ref{result-infinit-osc}
are proved by a careful analysis of certain energy levels
associated to (P).


\section{A key result}

For a fixed $c>0$, we consider the problem $$
\left\{\begin{array}{lll} -\Delta(\Delta u(k-1))+cu(k)=g(u(k)),
\quad k\in[1,T],\\ u(0)=u(T+1)=0,
 \end{array}\right.\eqno{\rm (P}_c)$$
 where $g:\mathbb R\to \mathbb R$ is a continuous function.  Moreover, let $E_c:X\to \mathbb R$ be
 the energy functional associated to problem ${\rm (P}_c)$ defined by
 $$E_c(u)=\frac{1}{2}\|u\|^2+\frac{c}{2}\sum_{k=1}^T (u(k))^2-\mathcal G(u), \ \ u\in X,$$
 where
 $$\mathcal G(u)=\sum_{k=1}^T G(u(k)), \ \ {\rm and} \ \ G(s)=\int_0^sg(t)dt,\ \ s\in \mathbb R.$$
It is immediate to show that $E_c$  is well-defined, it belongs to
$C^1(X;\mathbb R)$ and $$E_c'(u)(v)=\langle u,v\rangle
+c\sum_{k=1}^T u(k)v(k)-\sum_{k=1}^T g(u(k))v(k),\ \ \forall
u,v\in X.$$ Since we have $$\langle
u,v\rangle=-\sum_{k=1}^{T+1}\Delta(\Delta u(k-1))v(k),$$ an
element  $u\in X$ is a solution for ${\rm (P}_c)$ if
$E_c'(u)(v)=0$ for every $v\in X$, i.e., $u$ is a critical point
of $E_c$.

 Let $d<0<a<b$ some fixed numbers. We
introduce the set
\begin{equation}\label{halmaz-ertelmezes}
  N^b=\{u\in X:d\leq u(k)\leq
b\ \ {\rm for\ every}\ \ k\in \mathbb [1,T]\}.
\end{equation}
 We assume on $g:\mathbf R \to \mathbf R$ that
\begin{enumerate}
\item[$(H_g)$] \ \ \ $g(s)=0$ for $s\leq 0$, and $g(s)\leq 0$ for every $s\in [a,b].$
\end{enumerate}
The main result of this section is as follows.

\begin{proposition}\label{prop-weak} Assume that $g:\mathbf R \to \mathbf R$ verifies $(H_g).$  Then
\begin{enumerate}
\item[(a)] $E_c$ is bounded from below on
$N^b$ attaining its infimum at $\tilde u\in N^b$;

\item[(b)] $\tilde u(k)\in [0,a]$ for every $k\in
[1,T]$;

\item[(c)]  $\tilde u$ is a solution of ${\rm (P}_c)$.
\end{enumerate}
\end{proposition}


{\it Proof.}  $(a)$ Since the norms $\|\cdot\|_\infty$ and
$\|\cdot\|$ are equivalent in the finite-dimensional space $X$,
the set $N^b$ is compact in $X.$ Combining this fact with the
continuity of $E_c$, we  infer that $E_c|_{N^b}$ attains its
infimum at $\tilde u\in N^b$.

$(b)$   Let $K=\{k\in \mathbb [1,T]:\tilde u(k)\notin [0,a]\}$ and
suppose that $K\neq \emptyset.$ Define the truncation function
$\gamma:\mathbf R\to \mathbf R$ by $\gamma(s)=\min(s_+,a)$, where
$s_+=\max(s,0).$ Now, set $w=\gamma\circ \tilde u.$ Since
$\gamma(0)=0$ we have that $w(0)=w(T+1)=0$, so $w\in X$. Moreover,
$w(k)\in [0,a]$ for every $k\in [1,T]$; thus  $w\in N^a\subset
N^b.$

 We introduce the sets $$K_-=\{ k\in K :
\
 \tilde u(k)<0\} \qquad \mbox{and} \qquad K_+= \{k\in K : \
\tilde u(k)>a\}.$$ Thus, $K=K_-\cup K_+,$ and we have that
$w(k)=\tilde u(k)$ for all $k\in [1,T]\setminus K$, $w(k)=0$ for
all $k\in K_-$, and $w(k)=a$ for all $k\in K_+$. Moreover,
 we have
\begin{eqnarray}\label{elso-I-k}
 E_c(w) -E_c(\tilde u)&=&\frac{1}{2}(\|w\|^2-\|\tilde u\|^2)+\frac{c}{2}\sum_{k=1}^T [(w(k))^2-(\tilde u(k))^2]-[\mathcal G(w)-\mathcal G(\tilde
 u)]\nonumber \\&=:&\frac{1}{2}I_1+\frac{c}{2}I_2-I_3.
 \end{eqnarray}
Since $\gamma$ is a Lipschitz function with Lipschitz-constant 1,
and $w=\gamma \circ \tilde u$, we have
\begin{eqnarray}\label{I1}
  I_1&=&\|w\|^2-\|\tilde u\|^2=\sum_{k=1}^{T+1}[|\Delta
w(k-1)|^2-|\Delta \tilde
u(k-1)|^2]\nonumber\\&=&\sum_{k=1}^{T+1}[|w(k)-w(k-1)|^2-| \tilde
u(k)-\tilde u(k-1)|^2]\nonumber\\&\leq& 0.
\end{eqnarray}
Moreover, we have
\begin{eqnarray}\label{I2}
  I_2&=&\sum_{k=1}^T [(w(k))^2-(\tilde u(k))^2]=\sum_{k\in K}[(w(k))^2-(\tilde u(k))^2]\nonumber
  \\&=&\sum_{k\in K_-}-(\tilde u(k))^2+\sum_{k\in K_+}[a^2-(\tilde u(k))^2]\nonumber\\&\leq& 0.
\end{eqnarray}
Next, we estimate $I_3$. First, let us point out that $G(s)=0$ for
$s\leq 0$; thus, $\sum_{k\in K_-}[G(w(k))-G(\tilde u(k))]=0.$ By
the mean value theorem, for every $k\in K_+$, there exists $n_k\in
[a,\tilde u(k)]\subset [a,b]$ such that $G(w(k))-G( \tilde
u(k))=G(a)-G( \tilde u(k))=g(n_k)( a-\tilde u(k)).$ Taking into
account hypothesis $(H_g)$, we have that $G(w(k))-G( \tilde
u(k))\geq 0$ for every $k\in K_+$. Consequently,
\begin{eqnarray}\label{I3}
  I_3&=&\mathcal G(w)-\mathcal G(\tilde u)=\sum_{k\in K}[G(w(k))-G(\tilde
  u(k))]=\sum_{k\in K_+}[G(w(k))-G(\tilde
  u(k))]\nonumber\\&\geq& 0.
\end{eqnarray}
Combining relations (\ref{I1})-(\ref{I3}) with (\ref{elso-I-k}),
we have that
 $$E_c(w) -E_c(\tilde u)\leq 0.$$
On the other hand, since $w\in N^b$, then  $E_c(w)\geq E_c(\tilde
u)=\inf_{N^b}E_c$. So, every term in $E_c(w) -E_c(\tilde u)$
should be zero.
In particular, from $I_2$, we have  %$$\int_W|\nabla \tilde
%u_k|^p=0,$$
$$\sum_{k\in K_-}(\tilde u(k))^2=\sum_{k\in K_+}[a^2-(\tilde
u(k))^2]=0,$$ which imply that $\tilde u(k)=0$ for every $k\in
K_-$ and $\tilde u(k)=a$ for every $k\in K_+$. By definition of
the sets $K_-$ and $K_+$, we must have $K_-=K_+=\emptyset$, which
contradicts $K_-\cup K_+=K\neq \emptyset$.


$(c)$ Let us fix $v\in X$ arbitrarily. Due to $(b)$, it is clear
that $\tilde u+\varepsilon v\in N^b$ for $|\varepsilon|$ small
enough. Consequently, due to $(a)$, the function
$j(\varepsilon)=E_c(\tilde u+ \varepsilon v)$ has its minimum at
$0$; being differentiable at $0$, we have that $j'(0)=0,$ i.e.,
 $E_c'(\tilde u)(v)=0,$ which means that $\tilde u$ is a solution of $({\rm P}_c).$ This completes the proof. \hfill
$\square$

\section{Proof of Theorem \ref{result-zero-osc}}

We assume hypothesis $(H^0)$ holds. In particular, we have
$f(0)=0$. One may fix $c_0>0$ such that
%\begin{equation}\label{H_0-version}
$   \liminf_{s\to 0^+}\frac{f(s)}{s}<-c_0<0.$
%\end{equation}
 Consequently, there is a  sequence $\{\overline
s_n\}_n\subset (0,1)$
 converging (decreasingly) to 0, such that
\begin{equation}\label{f(s_k)<c_0}
  f(\overline s_n)<-c_0 \overline  s_n.
\end{equation}
 Let us define the functions $g_0,G_0:\mathbb R\to \mathbb R$ by\\
\begin{equation}\label{G-definit}
  g_0(s)=f(s_+)+c_0s_+\ \ \ {\rm and}\ \ \
G_0(s)=\int_0^s g_0(t)dt,\ \ s\in \mathbb R,
\end{equation}
where $s_+=\max(s,0).$ Due to (\ref{f(s_k)<c_0}),  $g_0(\overline
s_n)<0$; so, there are two sequences $\{a_n\}_n$,
$\{b_n\}_n\subset (0,1)$, both converging to 0, such that
$b_{n+1}<a_n<\overline s_n<b_n$ for every $n\in \mathbb N$ and $$
g_0(s)\leq  0\ {\rm for\ all}\ s\in [a_n,b_n]. $$ In this way,
hypothesis $(H_g)$ is verified for $g_0$ on every interval
$[a_n,b_n]$, $n\in \mathbb N$. Applying Proposition
\ref{prop-weak} to every interval $[a_n,b_n],n\in \mathbb N,$ the
problem $$ \left\{\begin{array}{lll} -\Delta(\Delta
u(k-1))+c_0u(k)=g_0(u(k)), \quad k\in[1,T],\\ u(0)=u(T+1)=0,
 \end{array}\right.\eqno{\rm (P}_{c_0})$$ has a sequence of non-negative solutions
$\{u_n^0\}_n\subset X$, where $u_n^0$ is a relative minimum of the
functional  $E_{c_0}$ associated to ${\rm (P}_{c_0})$ on the set
$N^{b_n}$, $n\in \mathbb N.$ Furthermore, since $g_0(s)=f(s)+c_0s$
on the interval $(0,1)$, the elements $u_n^0$ are also solutions
of problem ${\rm (P)}$. Moreover, due to Proposition
\ref{prop-weak} $(b)$, we also have
\begin{equation}\label{utildeintervall}
0\leq u_n^0(k)\leq a_n \ \ {\rm for \ every}\ k\in [1,T],\  n\in
\mathbb N.
\end{equation}


In the sequel, carrying out an energy-level analysis, we prove
that there are infinitely many distinct elements in the sequence
$\{u_n^0\}_n\subset X$. Due to $(H^0)$ and (\ref{G-definit}), we
have that $\limsup_{s\to
0^+}\frac{G_0(s)}{s^2}>\frac{1}{T}+\frac{c_0}{2}.$ In particular,
there exists a sequence $\{s_n\}_n$ with $0<s_n\leq a_n$, $n\in
\mathbb N,$ and
$$G_0(s_n)>\left(\frac{1}{T}+\frac{c_0}{2}\right)s_n^2.$$ Define
the function $w_n\in X$ by $w_n(k)=s_n$ for every $k\in [1,T]$.
Then, we have
\begin{eqnarray*}
   E_{c_0}(w_n)&=&\frac{1}{2}\sum_{k=1}^{T+1}|\Delta
w_n(k-1)|^2+\frac{c_0}{2}\sum_{k=1}^T (w_n(k))^2- \sum_{k=1}^T
G_0(w_n(k))\\ &=& s_n^2+\frac{c_0T}{2}s_n^2-T G_0(s_n)
\\&<& s_n^2+\frac{c_0T}{2}s_n^2-T\left(\frac{1}{T}+\frac{c_0}{2}\right)s_n^2
\\&=&0.
\end{eqnarray*}
The above estimation and $w_n\in N^{s_n}\subset N^{b_n}$ show that
\begin{equation}\label{3A}
E_{c_0}(u_n^0)=\min_{N^{b_n}} E_{c_0}\leq E_{c_0}(w_n)<0\ \ {\rm
for\ all}\ n\in \mathbb N.
\end{equation}
Once we prove that
\begin{equation}\label{3B}
  \lim_{n\to \infty}E_{c_0}(u_n^0)=0,
\end{equation}
our claim holds.  Indeed, (\ref{3A}) and (\ref{3B}) imply that
there are infinitely many distinct elements in the sequence
$\{u_n^0\}_n\subset X$. We clearly have
\begin{eqnarray*}
   E_{c_0}(u_n^0)&=&\frac{1}{2}\sum_{k=1}^{T+1}|\Delta
u_n^0(k-1)|^2+\frac{c_0}{2}\sum_{k=1}^T (u_n^0(k))^2- \sum_{k=1}^T
G_0(u_n^0(k))\\ &\geq & -\sum_{k=1}^T G_0(u_n^0(k))\geq
-\sum_{k=1}^T u_n^0(k)\max_{s\in [0,u_n^0(k)]}|g_0(s)|
\\&\geq &-\max_{s\in [0,a_n]}|g_0(s)|\sum_{k=1}^T u_n^0(k)
\\&\geq&-a_n T \max_{s\in [0,1]}|g_0(s)|.
\end{eqnarray*}
Since $\lim_{n\to \infty}a_n=0$, the above estimate and (\ref{3A})
yield (\ref{3B}).

Relation (\ref{elsobecslesnullaban}) is an immediate consequence
of (\ref{utildeintervall}), $\lim_{n\to \infty}a_n=0$, and to the
fact that the norms $\|\cdot\|_\infty$ and $\|\cdot\|$ are
equivalent. The proof of Theorem \ref{result-zero-osc} is
complete.



\section{Proof of Theorem \ref{result-infinit-osc}}

The proof is similar to that of Theorem \ref{result-zero-osc}.  We
assume hypothesis $(H^\infty)$ holds.  We choose $c_\infty>0$ such
that
%\begin{equation}\label{H_0-version}
$   \liminf_{s\to \infty}\frac{f(s)}{s}<-c_\infty<0.$
%\end{equation}
Consequently, we may fix a sequence $\{\overline s_n\}_n\subset
(0,\infty)$ such that $\lim_{n\to \infty}\overline s_n
=\infty$ and
\begin{equation}\label{f(s_k)<c_0-veg}
  f(\overline s_n)<-c_\infty \overline  s_n.
\end{equation}
 We define the functions $g_\infty,G_\infty:\mathbb R\to \mathbb R$ by\\
\begin{equation}\label{G-definit-veg}
  g_\infty(s)=f(s_+)+c_\infty s_+\ \ \ {\rm and}\ \ \
G_\infty(s)=\int_0^s g_\infty(t)dt,\ \ s\in \mathbb R.
\end{equation}
 Due to the right hand side inequality of $(H^\infty)$ and
(\ref{G-definit-veg}), we have that $\limsup_{s\to
\infty}\frac{G_\infty(s)}{s^2}>\frac{1}{T}+\frac{c_\infty}{2}.$ In
particular, for a small $\varepsilon_\infty>0,$ there exists a
sequence $\{s_n\}_n$ %with $0<s_n\leq a_n$, $n\in \mathbb N,$
tending to $\infty$ such that
\begin{equation}\label{G-infty-becsles}
  G_\infty(s_n)>\left(\frac{1}{T}+\frac{c_\infty}{2}+\varepsilon_\infty\right)s_n^2.
\end{equation}
Since $\lim_{n\to \infty} \overline s_n=\infty,$ one can fix a
subsequence $\{\overline s_{m_n}\}_n$ of $\{\overline s_{n}\}_n$
such that $s_n \leq \overline s_{m_n}$ for every $n\in \mathbf N.$
On account of (\ref{f(s_k)<c_0-veg}),  $g_\infty(\overline
s_{m_n})<0$; thus, we may fix two sequences $\{a_n\}_n$,
$\{b_n\}_n\subset (0,\infty)$ such that   $a_n<\overline
s_{m_n}<b_n<a_{n+1}$ for every $n\in \mathbb N$, $\lim_{n\to
\infty}a_n=\lim_{n\to \infty}b_n=\infty$, and $$ g_\infty(s)\leq
0\ {\rm for\ all}\ s\in [a_n,b_n]. $$ Consequently, the function
$g_\infty$ fulfills $(H_g)$  on every interval $[a_n,b_n]$, $n\in
\mathbb N$. We apply Proposition \ref{prop-weak} to every interval
$[a_n,b_n],n\in \mathbb N,$ obtaining that the problem $$
\left\{\begin{array}{lll} -\Delta(\Delta u(k-1))+c_\infty
u(k)=g_\infty (u(k)), \quad k\in[1,T],\\ u(0)=u(T+1)=0,
 \end{array}\right.\eqno{\rm (P}_{c_\infty})$$ has a sequence of non-negative solutions
$\{u_n^\infty\}_n\subset X$, where $u_n^\infty$ is a relative
minimum of the functional  $E_{c_\infty}$ associated to ${\rm
(P}_{c_\infty})$ on the set $N^{b_n}$, $n\in \mathbb N.$ Since
$g_\infty(s)=f(s)+c_\infty s$ on $[0,\infty)$, the elements
$u_n^\infty$ are solutions not only for ${\rm (P}_{c_\infty})$ but also for ${\rm (P)}$. %Furthermore, we
%have that
%\begin{equation}\label{utildeintervall-veg}
%0\leq u_n^\infty(k)\leq a_n \ \ {\rm for \ every}\ k\in [1,T],\
%n\in \mathbb N.
%\end{equation}


Now, we are going to prove that there are infinitely many distinct
elements in the sequence $\{u_n^\infty\}_n\subset X$. To do this,
it is enough to show that
\begin{equation}\label{3B-veg}
  \lim_{n\to \infty}E_{c_\infty}(u_n^\infty)=-\infty.
\end{equation}
Define the function $w_n\in X$ by $w_n(k)=s_n$ for every $k\in
[1,T]$. Then, by using (\ref{G-infty-becsles}), we have
\begin{eqnarray*}
   E_{c_\infty}(w_n)&=&\frac{1}{2}\sum_{k=1}^{T+1}|\Delta
w_n(k-1)|^2+\frac{c_\infty}{2}\sum_{k=1}^T (w_n(k))^2-
\sum_{k=1}^T G_\infty(w_n(k))\\ &=& s_n^2+\frac{c_\infty
T}{2}s_n^2-T G_\infty(s_n)
\\&<& s_n^2+\frac{c_\infty T}{2}s_n^2-T\left(\frac{1}{T}+\frac{c_\infty}{2}+\varepsilon_\infty\right)s_n^2
\\&=&-T\varepsilon_\infty s_n^2.
\end{eqnarray*}
By construction, we know that $w_n\in N^{s_n}\subset N^{b_n}$,
thus
\begin{equation}\label{3A-veg}
E_{c_\infty}(u_n^\infty)=\min_{N^{b_n}} E_{c_\infty}\leq
E_{c_\infty}(w_n)<-T\varepsilon_\infty s_n^2\ \ {\rm for\ all}\
n\in \mathbb N.
\end{equation}
Since $\lim_{n\to \infty}s_n=\infty,$ relation (\ref{3A-veg})
implies  (\ref{3B-veg}).

It remains to prove (\ref{vegtelen-asszimpt}). Since the norms
$\|\cdot\|_\infty$ and $\|\cdot\|$ are equivalent, it is enough to
prove the former limit, i.e., $\lim_{n\to \infty}\|
u_n^\infty\|_{\infty}=\infty.$ By contradiction, we assume that
for a subsequence %$\{u_{m_n}^\infty\}_n$
of $\{u_n^\infty\}_n$, still denoted by $\{u_n^\infty\}_n$, one
can find a constant  $C>0$ such that $\|u_{n}^\infty\|_\infty\leq
C$ for every $n\in \mathbb N$. Therefore, we have
$$E_{c_\infty}(u_n^\infty)\geq - \sum_{k=1}^T
G_\infty(u_n^\infty(k))\geq -T \max_{s\in [0,C]} |G_\infty(s)|\ \
{\rm for\ every}\ n\in \mathbb N.$$ This inequality contradicts
relation (\ref{3B-veg}) which completes the proof of Theorem
\ref{result-infinit-osc}.\\
\begin{remark}
When $T=2$, the conclusions of Theorems \ref{result-zero-osc} and
\ref{result-infinit-osc} may be obtained in a very simple way. In
this case, it is enough to solve the system $$
\left\{\begin{array}{lll} 2 a-b=f(a), \\ 2 b-a=f(b),\\ a,b>0.
 \end{array}\right.\eqno{\rm (P')}$$
\end{remark}
Indeed, a solution of (P) is any function $u:[0,3]\to\mathbb R$
defined by $u(0)=u(3)=0$, $u(1)=a$, $u(2)=b.$ As one can observe,
if there is a sequence of distinct fixed points for $f$, say
$\{c_n\}_n\subset  (0,\infty)$, we have infinitely many solutions
for problem ${\rm (P')}$ of the form $(a,b)=(c_n,c_n).$ Let us
assume the contrary, i.e., there is at most finite number of
distinct fixed points for $f$. Combining this assumption with the
left hand side of $(H^0)$, there exists a $\delta>0$ such that
$f(s)<s$ for every $s\in (0,\delta)$. After an integration we
obtain that $$\limsup_{s\to 0^+}\frac{F(s)}{s^2}\leq \frac{1}{2}=
\frac{1}{T}$$ which contradicts the right hand side of $(H^0)$. In
a similar manner, when $(H^\infty)$ holds, we can fix a compact
set $L\subset [0,\infty)$  such that $f(s)<s$ for every $s\in
(0,\infty)\setminus L$,  which contradicts the right hand side of
$(H^\infty)$.

 The above arguments also suggest that the constant $\frac{1}{T}$ in $(H^0)$ and $(H^\infty)$ is optimal.
\\


\noindent {\bf Acknowledgments.} A. Krist\'aly is supported by
Grant CNCSIS, IDEI 527/2007.  M. Mih\u ailescu  and V. R\u
adulescu are supported by Grant CNCSIS, IDEI 79/2007.

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