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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 192, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/192\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions for nonlinear discrete inclusions}

\author[N. Marcu, G. Molica Bisci\hfil EJDE-2012/192\hfilneg]
{Nicu Marcu, Giovanni Molica Bisci}  % in alphabetical order

\address{Nicu Marcu \newline
Department of Finance, University of Craiova,
Street A.I. Cuza No. 13, 200585, Craiova, Romania}
\email{marcu.nicu@yahoo.com}

\address{Giovanni Molica Bisci \newline
Dipartimento MECMAT, University of Reggio Calabria,
Via Graziella, Feo di Vito, 89124 Reggio Calabria, Italy}
\email{gmolica@unirc.it}

\thanks{Submitted September 4, 2012. Published November 6, 2012.}
\subjclass[2000]{38J20, 39A10, 34B15, 49J40}
\keywords{Nonlinear algebraic systems; difference equations;\hfill\break\indent
 non-smooth problems; variational methods}

\begin{abstract}
 A non-smooth abstract result is used for proving the existence of at
 least one nontrivial solution of an algebraic discrete inclusion.
 Successively, a multiplicity theorem for the same class of discrete
 problems is also established by using a locally Lipschitz continuous
 version of the famous Br\'{e}zis-Nirenberg theoretical result in
 presence of splitting. Some applications to tridiagonal, fourth-order
 and partial difference inclusions are pointed out.
\end{abstract}

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

\section{Introduction}

 A considerable number of problems, which are strictly connected both with 
boundary value differential problems and numerical simulations of some
 mathematical models arising from many research areas 
(biological, physical and computer science) can be formulated as special cases 
of nonlinear algebraic systems (see, for instance \cite{ZB}).

In this article, motivated by this large interest, we investigate the
 existence of solutions for discrete algebraic inclusions. 
More precisely, let $T>1$ be a positive integer and let $g_k:\mathbb{R}\to \mathbb{R}$
be a locally essentially bounded function,
for every $k\in {\mathbb{Z}}[1,T]:=\{1,2,\dots ,T\}$. We are interested either 
on the existence or in multiple solutions for the  discrete inclusion
\begin{equation}  \label{W} %\tag{$S_{A}^{g}$}
\sum_{l=1}^{T}a_{kl}u_l\in [g^-_k(u_k),g^+_k(u_k)],
\quad(\forall\,k\in {\mathbb{Z}}[1,T]),
\end{equation}
where $A:=(a_{ij})_{T\times T}$ is a real symmetric positive definite matrix and
$$
g^{-}_k(t):=\lim_{\delta\to 0^{+}}\operatorname{ess\,inf}_{|\xi-t|<\delta}g_k(\xi),\quad
g^{+}_k(t):=\lim_{\delta\to 0^{+}}\operatorname{ess\,sup}_{|\xi-t|<\delta}g_k(\xi),
$$
for every $k\in {\mathbb{Z}}[1,T]$.

 It is clear that if the functions $g_k$ are continuous
(instead of locally essentially bounded) problem \eqref{W} becomes a more 
familiar nonlinear algebraic system
$$
Au= g(u),
$$
 in which $u=(u_1,\dots ,u_T)^{t}\in\mathbb{R}^T$ and
$g(u):=(g_1(u_1),\dots ,g_T(u_T))^t$.

However, to the best of our knowledge, for discrete difference inclusions
 there are only few papers involving the second-order difference operator.
 For instance, in \cite{AOL}, the existence of at least
one solution was obtained via the set-valued mapping theory,
 while in \cite{Z45}, existence results for suitable second-order 
discrete discontinuous equations have been investigated by variational methods. 
The aim of this paper is to establish existence and multiplicity results 
for algebraic discrete inclusions like problem \eqref{W}.

The main existence result contained here (see Theorem \ref{Main0}) is 
obtained using a non-smooth critical points theorem contained 
in \cite[Theorem 2.1; part (a)]{BoMo}. This theoretical argument
 represents a non-smooth refinement of the quoted variational principle 
of Ricceri (see \cite{Ricceri}).

 Through this variational approach, we are able to prove the existence of 
one solution for problem \eqref{W}
just requiring that there is a real constant $\bar \gamma>0$ such that
$$
\frac{\bar\gamma^2}{ \sum_{k=1}^{T} 
\max_{|\xi|\leq \bar\gamma}\int_0^{\xi}g_k(t)dt}>\frac{2}{\lambda_1},
$$
where $\lambda_1$ is the 1-th eigenvalue of the symmetric and positive definite 
matrix $A$ (see Theorem \ref{Main0} as well as Remarks \ref{Y1} and \ref{Y2}).

Successively, a two solutions result for the algebraic inclusion \eqref{W} 
is proved (see Theorem \ref{Main} and Corollary \ref{Main2}).
Our proof in this case is based on an extension of the famous 
Br\'{e}zis-Nirenberg result \cite[Theorem 4]{BN}
obtained by Wu in \cite[Theorem 2.3]{Wu} for locally Lipschitz continuous functionals.

Due to the generality of \eqref{W}, remarkable applications are easily achieved.
Indeed, Theorem \ref{Main} can be used proving either existence or multiplicity 
of solutions for discrete inclusions involving certain tridiagonal matrices, 
fourth-order  discrete problems and partial difference inclusions 
(see Example \ref{esempiotto} and Section 5).

A special case of Theorem \ref{Main0} reads as follows.

\begin{theorem}\label{Main01}
Let $g_k:\mathbb{R}\to  \mathbb{R}$ be a locally essentially bounded and nonnegative function,
for every $k\in {\mathbb{Z}}[1,T]$. Assume that
$$
\limsup_{\gamma\to  +\infty}\frac{\gamma^2}{ \sum_{k=1}^{T}\int_0^{\gamma}g_k(t)dt}
=+\infty,
$$
and $g_k^{-}(0)>0$, for some $k\in {\mathbb{Z}}[1,T]$.
Then problem \eqref{W} admits at least one nontrivial solution.
\end{theorem}

Moreover, denoting by $\lambda_\ell^{(4)}$ and $\lambda_{\ell+1}^{(4)}$ 
respectively the $\ell$-th and $(\ell+1)$-th eigenvalue of the discrete problem
\begin{gather*}
\Delta^4u_{k-2}=\lambda u_k,\quad(\forall\, k  \in
\mathbb{Z}[1,T]) \\
u_{-2}=u_{-1}=u_0=0\\
u_{T+1}=u_{T+2}=u_{T+3}=0,
\end{gather*}
 one has the following multiplicity property.

\begin{theorem}\label{Introduction}
Let $h:\mathbb{R}\to\mathbb{R}$ be locally essentially bounded positive function and
consider the usual forward difference operator $\Delta u_{k-1}:=u_{k}-u_{k-1}$.
Assume that
\begin{itemize}
\item[(H1)] $ \limsup_{|t|\to  \infty}\frac{ h(t)}{t}=0$;
\item[(H2)] There exists an integer $\ell\in {\mathbb{Z}}[1,T-1]$ such that
 $$  
\frac{\lambda_{\ell}^{(4)}}{T}\leq \lim_{t\to  0}\frac{  h(t)}{t}
\leq\frac{\lambda_{\ell+1}^{(4)}}{T},\quad (\forall\, k\in {\mathbb{Z}}[1,T])
$$
\end{itemize}
Then the  forth-order discrete inclusion
\begin{equation} \label{4order} %\tag{$D_{h}$} 
\begin{gathered}
\Delta^4u_{k-2}\in [h^-(u_k),h^+(u_k)],\quad (\forall\,k\in {\mathbb{Z}}[1,T])\\
{u_{-2}=u_{-1}=u_0=0}\\
u_{T+1}=u_{T+2}=u_{T+3}=0.\\
\end{gathered}
\end{equation}
 admits at least two nontrivial solutions.
\end{theorem}

We emphasize that our results are also new in the continuous setting. 
In this case, the existence and multiplicity of solutions was investigated 
in a large number of other papers under various assumptions (see
for instance \cite{Z,ZB,ZC,ZF,Z45} and references therein). 
See also the recent papers \cite{CMolica, KMR, KMRT, MRT} for related topics.

The plan of the paper is as follows. In the next section we introduce our 
abstract framework. Successively, in sections 3 and 4, we show our existence 
and multiplicity results. A concrete example of an application of our 
abstract results to discrete partial inclusions is presented in the last section.

\section{Basic definitions and preliminary results}

 Let $(X,\Vert\cdot\Vert)$ be a real Banach space.
We denote by $X^*$ the dual space of $X$, while
$\langle\cdot,\cdot\rangle$ stands for the duality pairing between
$X^*$ and $X$.

 A function $J:X\to\mathbb{R}$ is called locally
Lipschitz continuous if to every $x\in X$ there corresponds a
neighborhood $V_x$ of $x$ and a constant $L_x\geq 0$ such that
$$
\vert J(z)-J(w)\vert\leq L_x\Vert z-w\Vert,\quad(\forall\, z,w\in V_x)\,.
$$
If $x,z\in X$, we write $J^{0}(x;z)$ for the generalized
directional derivative of $J$ at the point $x$ along the
direction $z$; i.e.,
$$
J^{0}(x;z):=\limsup_{w\to x,\, t\to 0^+} \frac{J(w+tz)-J(w)}{t}\,.
$$
The generalized gradient of the function $J$ in $x$,
denoted by $\partial J(x)$, is the set
$$
\partial J(x):=\{ x^*\in X^*: \langle x^*,z\rangle\leq J^{0}(x;z),
\;\forall\, z\in X\}.
$$
The basic properties of generalized directional derivative and generalized 
gradient were studied in \cite{CHANG,CLARKE2}.

We recall that if $J$ is continuously G\^{a}teaux differentiable at $u$, then $J$ is locally
Lipschitz at $u$ and $\partial J(u) = \{J '(u)\}$, where $J'(u)$ stands for the first derivative of $J$ at $u$.
Further, a point $u$ is called a (generalized)
critical point of the locally Lipschitz continuous function $J$ if
 $0_{X^*}\in \partial J(u)$; i.e.,
$$
J^0(u;z)\geq 0,
$$
 for every $z\in X$.
Clearly, if $J$ is a continuously G\^{a}teaux differentiable at $u$, then $u$
becomes a (classical) critical point of $J$, that is $J '(u) = 0_{X^*}$.

A locally Lipschitz functional $J:X\to \mathbb{R}$ is said to fulfill the Palais-Smale
condition if
\begin{itemize}
\item[(PS)]  Every sequence
$\{x_n\}\subset X$ such that $\{J(x_n)\}$ is bounded and
$$
J^0(x_n;x-x_n)\geq -\varepsilon_n\|x-x_n\|,\quad (\forall\,x\in X)
$$
where $\varepsilon_n\to  0^+$,
possesses a convergent subsequence.
\end{itemize}
For an complete overview on the non-smooth calculus we mention the
monograph \cite{MR}. Further, we cite a very recent book \cite{KRV} 
as a general reference on this subject.

Our main tool will be the following two abstract critical point theorems, 
for locally Lipschitz continuous functions.

\begin{theorem}[{\cite[Theorem 2.1; part (a)]{BoMo}}] \label{BMB}
Let $X$ be a reflexive real Banach space, and let $\Phi,\Psi:X\to\mathbb{R}$
be two locally Lipschitz continuous functionals such that $\Phi$ is
sequentially weakly lower semicontinuous and coercive and $\Psi$
is sequentially weakly upper semicontinuous. For every $\rho>\inf_X
\Phi$, put
$$
\varphi(\rho):=\inf_{u\in\Phi^{-1}(]-\infty,\rho[)}
\frac{  \sup_{v\in\Phi^{-1}(]-\infty,\rho[)}\Psi(v)-\Psi(u)}{\rho-\Phi(u)}.
$$
Then, for every $\rho>\inf_X\Phi$ and every
$\lambda\in ]0,{1}/{\varphi(\rho)}[$, the restriction
of the functional $J_\lambda:=\Phi-\lambda\Psi$ to
$\Phi^{-1}(]-\infty,\rho[)$ admits a global minimum, which is a
critical point $($local minimum$)$ of $J_\lambda$ in $X$.
\end{theorem}

\begin{theorem}\cite[Theorem 2.3]{Wu}\label{splitting}
Suppose that $X:=X_1\oplus X_2\,$ with ${\rm dim}(X_1)>0$ and
 $0<{\rm dim}(X_2)<\infty$. Let $J$ be a locally Lipschitz 
 continuous functional satisfying the $({\rm PS})$ condition and such that
\begin{gather*}
J(u)\leq 0,\quad(\forall\, u\in\bar{B}({0,\rho})\cap X_2),\\
J(u)\geq 0,\quad(\forall\, u\in\bar{B}({0,\rho})\cap X_1)
\end{gather*}
for some $\rho>0$.
 Assume also that $J$ is bounded from below and $ \inf_{u\in X}J(u)<0$.
 Then $J$ has at least two nonzero critical points.
\end{theorem}

\begin{remark} \label{rmk2.1}\rm
As pointed out in Introduction Theorem \ref{BMB} can be view as a non-smooth
 version of the quoted variational principle of Ricceri; see the paper \cite{Ricceri}. 
Further, Theorem \ref{splitting} represents an extension, to the case of 
locally Lipschitz continuous functionals, of the celebrated critical point
theorem in presence of splitting established by Br\'{e}zis and
Nirenberg \cite[Theorem 4]{BN}. See for completeness the work \cite[Theorem 2.3]{Wu}.
\end{remark}

 Here, as the ambient space $X$, we consider the $T$-dimensional Banach space
 $\mathbb{R}^T$ endowed with the norm
$$
\|u\|:=\Big(\sum_{k=1}^{T}u_k^2\Big)^{1/2},
$$
induced by the standard Euclidean inner product 
$\langle u,v\rangle_X:= \sum_{k=1}^{T}u_kv_k$.

 Set ${\mathfrak{X}}_{T}$ to be the class of all symmetric and positive 
definite matrices of order $T$. Further, we denote by $\lambda_1,\dots ,\lambda_T$ 
the eigenvalues of $A$ (ordered as $0<\lambda_1\leq\dots \leq \lambda_T$) 
and by $\xi_1,\dots ,\xi_T$ the corresponding orthonormal eigenvectors.
It is well-known that if $A\in {\mathfrak{X}}_{T}$, for every $u\in X$, 
then one has
\begin{gather}\label{immersione2}
\lambda_1\|u\|^2\leq u^{t}Au\leq \lambda_T\|u\|^2, \\
\label{immersione}
\|u\|_\infty\leq \frac{1}{\sqrt{\lambda_1}}(u^{t}Au)^{1/2},
\end{gather}
 where $ \|u\|_{\infty}:= \max_{k\in {\mathbb{Z}}[1,T]}|u_k| $.

 For the rest of this article, we assume that $A\in{\mathfrak{X}}_{T}$.
For every $u\in X$, we put
$$
\Phi(u):=\frac{u^{t}Au}{2}, \quad  \Psi(u):=\sum_{k=1}^{T}G_k(u_k),  \quad 
J(u):=\Phi(u)-\Psi(u),
$$
 where
$G_k(t):= \int_{0}^{t}g_k(\xi)d \xi$, for every 
$(k,t)\in {\mathbb{Z}}[1,T]\times \mathbb{R}$.
It is easy to verify that $\Phi$ is continuously G\^{a}teaux differentiable,
 while $\Psi$ is locally Lipschitz continuous.

\begin{proposition}\label{equivalen}
Assume that $u\in X$ is a critical point of the functional $J$. 
Then $u$ is a solution of problem \eqref{W}.
\end{proposition}

\begin{proof}
If $u$ is a critical point of $J$, bearing in mind of \cite[Propositions 2.3.1 and
2.3.3]{CLARKE2}, it follows that
\begin{equation} \label{derivata}
\Phi'(u)(z)\leq \Psi^{0}(u;z)\leq \Big(\sum_{k=1}^{T}G^{0}_k(u_k;z_k)\Big),
\end{equation}
for every $z\in X$.
 Moreover,
\begin{equation} \label{rappresentazione}
\Phi'(u)(z)=\frac{\langle\nabla (u^{t}Au),z\rangle_{X}}{2},
\end{equation}
for every $z\in X$.

For every $\xi\in\mathbb{R}$ and $k\in {\mathbb{Z}}[1,T]$, 
by putting in \eqref{derivata} the vector $z=\xi e_k$,
where $e_k$ are the canonical unit vectors of $X$, and taking 
in mind \eqref{rappresentazione}, we obtain
$$
\langle \sum_{l=1}^{T}a_{kl}u_l,\xi\rangle_{\mathbb{R}}
=\Phi'(u)(z)\leq G^{0}_k(u_k;\xi),
$$
namely
$$
\sum_{l=1}^{T}a_{kl}u_l\in \partial G_k(u_k).
$$
 Finally, since it is well-known that
$$
\partial G_k(u_k)=[g^-_k(u_k),g^+_k(u_k)],
$$
for every $k\in {\mathbb{Z}}[1,T]$ (see for instance \cite[Example 2.2.5]{CLARKE2}) 
it follows that
$$
\sum_{l=1}^{T}a_{kl}u_l\in [g^-_k(u_k),g^+_k(u_k)],\quad
 (\forall\,k\in {\mathbb{Z}}[1,T]).
$$
 Therefore our assertion is proved.
\end{proof}


\section{A nontrivial solution}

The main result of this section reads as follows.
\begin{theorem}\label{Main0}
Let $g_k:\mathbb{R}\to  \mathbb{R}$ be a locally essentially bounded function, 
for every $k\in {\mathbb{Z}}[1,T]$. Assume that
\begin{equation}
\sup_{\gamma>0}\frac{\gamma^2}{ \sum_{k=1}^{T}\max_{|\xi|\leq \gamma}G_k(\xi)}
>\frac{2}{\lambda_1}.\label{SG}
\end{equation}
Then problem \eqref{W} admits at least one solution. Moreover if, in
 addition to the above condition, one has $g_k^{-}(0)>0$, for some
$k\in {\mathbb{Z}}[1,T]$, the obtained solution is nontrivial.
\end{theorem}

\begin{proof}
Since condition \eqref{SG} holds, there exists $\bar \gamma>0$ such that
\begin{equation}\label{ser1}
\frac{\bar\gamma^2}{ \sum_{k=1}^{T} \max_{|\xi|\leq \bar\gamma}G_k(\xi)}
>\frac{2}{\lambda_1}.
\end{equation}
Hence, take $\bar\rho:= \frac{\lambda_1\bar \gamma^2}{2}$ and apply 
Theorem \ref{BMB}.
Clearly, $ \inf_{u\in X}\Phi(u)<\bar\rho$ and
$$
\varphi(\bar\rho):=\inf_{u\in\Phi^{-1}(]-\infty,\bar\rho[)}
\frac{  \sup_{v\in\Phi^{-1}(]-\infty,\bar\rho[)}\Psi(v)
-\Psi(u)}{\bar\rho-\Phi(u)}
\leq \frac{  \sup_{v\in\Phi^{-1}(]-\infty,\bar\rho[)}\Psi(v)}{\bar\rho},
$$
taking into account that $0_X\in\Phi^{-1}(]-\infty,\bar\rho[)$ and 
$\Phi(0_X)=\Psi(0_X)=0$.

Now,  using condition \eqref{immersione}, it follows that
$$
\Phi^{-1}(]-\infty,\bar\rho[)\subseteq \{u\in X: \|u\|_{\infty} \le \gamma\}.
$$
Thus, the above remarks imply that
\begin{equation}\label{ser2}
\varphi(\bar\rho)\leq \frac{2}{\lambda_1}\frac{ \sum_{k=1}^{T}\max_{|\xi|
\leq \bar\gamma} G_k(\xi)}{\bar\gamma^2}.
\end{equation}
Consequently, by \eqref{ser1} and \eqref{ser2} one has $\varphi(\bar\rho)<1$. 
Hence, since $1\in\left]0,{1}/{\varphi(\bar\rho)}\right[$, 
Theorem \ref{BMB} ensures that the functional $J$ admits at least one critical
point $($local minima$)$ $\widetilde{u}\in \Phi^{-1}(]-\infty,\bar\rho[)$.

 Due to Proposition \ref{equivalen}, $\widetilde{u}\in X$ is a solution of
\eqref{W}. Under the additional assumption $g^-_k(0)>0$ 
(for some $k\in {\mathbb{Z}}[1,T]$) the obtained solution is clearly nontrivial.
\end{proof}

\begin{remark}\label{Y1}\rm
If in Theorem \ref{Main0} the functions $g_k$ are nonnegative, 
condition \eqref{SG} assumes the more simple and significative form
\begin{equation}
\sup_{\gamma>0}\frac{\gamma^2}{ \sum_{k=1}^{T}  G_k(\gamma)}
>\frac{2}{\lambda_1}.\label{SG'}
\end{equation}
Moreover, if
$$
\limsup_{\gamma\to  +\infty}\frac{\gamma^2}{ \sum_{k=1}^{T}  G_k(\gamma)}
>\frac{2}{\lambda_1},
$$
condition \eqref{SG'} automatically holds. Hence,
Theorem \ref{Main01} in Introduction is an immediate consequence
of Theorem \ref{Main0} taking into account the considerations above.
\end{remark}

\begin{remark}\label{Y2}\rm
Let $\bar \gamma>0$ be a real constant such that
$$
\frac{\bar\gamma^2}{ \sum_{k=1}^{T} \max_{|\xi|\leq \bar\gamma}G_k(\xi)}
>\frac{2}{\lambda_1},
$$
and said $\widetilde{u}\in X$ be the solution of problem \eqref{W} 
obtained by using Theorem \ref{Main0}. 
Hence, since $\widetilde{u}\in\Phi^{-1}(]-\infty,\bar\rho[)$, it follows that
$\|\widetilde{u}\|_\infty\leq \bar \gamma$.
\end{remark}

\begin{example}\label{esempiotto}\rm
Let $T\geq 3$ and $(a,b)\in \mathbb{R}^{-}\times\mathbb{R}^{+}$ be such that
 $$
\cos\big(\frac{\pi}{T+1}\big)<-\frac{b}{2a}.
$$
 Set
$$
{\operatorname{Trid}}_T(a,b,a)=\begin{pmatrix}
  b & a & 0 & \dots   & 0 \\
  a & b & a & \dots   & 0 \\
     &  & \ddots &  &  \\
  0 & \dots   & a & b & a \\
  0 & \dots   & 0 & a & b
\end{pmatrix}_{T\times T},
$$
 and consider the  discrete problem
\begin{equation}  \label{tridiagonalcase} %\tag{$S^{j}$}
L_{{\operatorname{Trid}}}(u)\in [j^-_k(u_k),j^+_k(u_k)],\quad (\forall\,k\in {\mathbb{Z}}[1,T])
\end{equation}
where
$$
L_{{\operatorname{Trid}}}(u):=\begin{cases}
bu_1+au_2\\
au_{k-1}+bu_k+au_{k+1},& (\forall\,k\in \{2,\dots ,T-1\})\\
au_{T-1}+bu_T,
\end{cases}
$$
and the functions $j_k:\mathbb{R}\to \mathbb{R}$ are assumed to be locally 
essentially bounded.
Hence, Theorem \ref{Main0} ensures that if
 $$
\sup_{\gamma>0}\frac{\gamma^2}{ \sum_{k=1}^{T}\max_{|\xi|\leq \gamma}
\int_0^\xi j_k(t)dt}>\frac{2}{  b+2a\cos\big(\frac{\pi}{T+1}\big)},
$$
problem \eqref{tridiagonalcase} admits one solution (see
\cite[Example 9; p.179]{Sc} for details). 
 Moreover if, in addition to our algebraic inequality, one also have 
$j_k^{-}(0)>0$, for some $k\in {\mathbb{Z}}[1,T]$, the obtained solution 
is nontrivial. The above result can be applied to second-order difference inclusions. 
Indeed, it is well-know that the  $T\times T$ matrix
$${\operatorname{Trid}}_T(-1,2,-1)
:=\begin{pmatrix}
  2 & -1 & 0 & \dots  & 0 \\
  -1 & 2 & -1 & \dots  & 0 \\
     &  & \ddots &  &  \\
  0 & \dots  & -1 & 2 & -1 \\
  0 & \dots  & 0 & -1 & 2
\end{pmatrix}
$$
in $\mathfrak{X}_{T}$, is associated to the second-order discrete
 boundary value problem
\begin{equation}  \label{dxi} %\tag{$\widetilde{S}^j$}
\begin{gathered}
-\Delta^2u_{k-1}\in[j^-_k(u_k),j^+_k(u_k)],\quad \forall k  \in
{\mathbb{Z}}[1,T] \\
{u_0=u_{T+1}=0,}
\end{gathered}
\end{equation}
where $\Delta^2u_{k-1}:=\Delta(\Delta u_{k-1})$, and, as usual,
 $\Delta u_{k-1}:=u_{k}-u_{k-1}$ denotes the forward difference operator.
\end{example}

\section{Two nontrivial solutions}

With the above notation and assumptions, the main result reads as follows.

\begin{theorem}\label{Main}
Assume that
\begin{itemize}
\item [(G1)] $ \limsup_{|\xi|\to  \infty}\frac{  G_k(\xi)}{\xi^2}<\frac{\lambda_1}{2}$,
for all $k\in {\mathbb{Z}}[1,T]$,
\end{itemize}
and that there exists an integer $\ell\in {\mathbb{Z}}[1,T-1]$ 
such that 
\begin{itemize}
\item[(G2)] $  \liminf_{\xi\to 0}\frac{  G_k(\xi)}{\xi^2}
\geq \frac{\lambda_{\ell}}{2}$,  for all $k\in {\mathbb{Z}}[1,T]$.
\end{itemize}
 Further, suppose that
\begin{itemize}
\item[(G3)] $  \limsup_{\xi\to 0}\frac{  G_k(\xi)}{\xi^2}
\leq \frac{\lambda_{\ell+1}}{2}$, for all $k\in {\mathbb{Z}}[1,T]$.
\end{itemize}
 Then problem \eqref{W} possesses at least two nontrivial
solutions.
\end{theorem}

\begin{proof}
 Our aim is to apply Theorem \ref{splitting}. From (G1), since $X$ 
is a finite dimensional space, it is easy to see that $J$ satisfies 
condition (PS).

 Indeed, using condition (G1), there are constants 
$\epsilon \in  ]0,{\lambda_1}/{2}[$ and
$\sigma>0$ such that
$$
{\frac{  G_k(\xi)}{\xi^{2}}}<\frac{\lambda_1}{2}-\epsilon,
$$
for every
$|t|\geq \sigma$ and $k\in {\mathbb{Z}}[1,T]$. Let us put
$$
M_1:=\max_{(k,\xi)\in {\mathbb{Z}}[1,T]\times [-\sigma,\sigma]}G_k(\xi).
$$
Therefore, for every $\xi\in\mathbb{R}$ and $k\in {\mathbb{Z}}[1,T]$, one has
$$
G_k(\xi)\leq M_1+M_2\xi^2,
$$
where
$$
M_2:=\frac{\lambda_1}{2}-\epsilon.
$$
 Moreover, the following inequality holds
$$
J(u)\geq \frac{u^{t}Au}{2}-\sum_{k=1}^{T}\Big[M_1+M_2u_k^2\Big],\quad 
(\forall\, u\in X).
$$
Hence
$$
J(u)\geq \frac{u^{t}Au}{2}-M_2\|u\|^2-TM_1,\,\,\ (\forall\, u\in X).
$$
 Thus, by using \eqref{immersione2}, one has
\begin{equation} \label{C}
J(u)\geq \epsilon \|u\|^2-TM_1,\quad (\forall\, u\in X)
\end{equation}
 which clearly shows that
$$
\lim_{\|u\|\to \infty}J(u)=+\infty.
$$
From this, and taking into account that $X$ is a finite $T$-dimensional
Hilbert space, it follows that the functional $J$ satisfies the (PS) condition.

We will prove now that, for some $\rho_1>0$, $J(u)\leq 0$ for every
 $u\in X_2\cap \bar{B}({0,\rho_1})$, where 
$X_2={\rm \overline{Span}}\{\xi_1, \ldots,\xi_{\ell}\}$. Thus, by condition
 (G2), there exists $\delta>0$ such that
$$
G_k(\xi)\geq \frac{\lambda_\ell}{2}\xi^2,\quad (\forall\, k\in {\mathbb{Z}}[1,T]),
$$
provided $0<|\xi|\leq\delta$.
Now, taking into account the discrete Cauchy-Schwarz inequality, one has
$$
|u_k|\leq \sum_{k=1}^{T}|u_k|\leq T^{1/2}\Big(\sum_{k=1}^{T}|u_k|^2\Big)^{1/2},
$$
for every $k\in {\mathbb{Z}}[1,T]$.
Then
$$
\|u\|_{\infty}\leq T^{1/2}\|u\|,\quad (\forall\, u\in X).
$$
Hence, for every $u\in\bar{B}({0,\rho_1})\cap X_2$, it follows that
$$
\|u\|_{\infty}\leq T^{1/2}\rho_1.
$$
 Consequently, if we take $\rho_1\leq\delta/T^{1/2}$, we obtain
\begin{equation}\label{tri}
G_k(u_k)\geq \frac{\lambda_{\ell}}{2}u_k^2,
\end{equation}
 for every $k\in{\mathbb{Z}}[1,T]$.
Moreover, if $u\in X_2$, there exists $a_k\in\mathbb{R}$ for every 
$k\in {\mathbb{Z}}[1,T]$, such that 
$$
u= \sum_{k=1}^{\ell}a_k\xi_k\quad\text{and}\quad
u^{t}Au= \sum_{k=1}^{\ell}\lambda_ka_k^2\leq \lambda_\ell\sum_{k=1}^{\ell}a_k^2.
$$
 Hence,
\begin{equation}\label{ttri}
u^{t}Au\leq   \lambda_{\ell}\|u\|^2,\,\,\,\,(\forall\, u\in X_2).
\end{equation}
Putting together (\ref{tri}) and (\ref{ttri}), we have
$$
J(u)\leq \frac{\lambda_{\ell}}{2}(\|u\|^2-\|u\|^2)=0,\quad
(\forall\, u\in \bar{B}({0,\rho_1})\cap X_2);
$$
that is,
$$
J(u)\leq 0,\quad(\forall\, u\in \bar{B}({0,\rho_1})\cap X_2).
$$
At this point, it remains to show that there exists $\rho_2>0$ such that
$$
J(u)\geq 0,\quad (\forall\, u\in\bar{B}({0,\rho_2})\cap X_1)
$$
where $X_1:= \overline{\operatorname{Span}}\{\xi_{\ell+1}, \dots,\xi_{T}\}$.

Fix $u\in X_1$, for suitable $b_k\in\mathbb{R}$ and for every 
$k\in {\mathbb{Z}}[\ell+1,T]$, one has that $u= \sum_{k=\ell+1}^{T}b_k\xi_k$ and
$$
J(u)\geq \sum_{k=\ell+1}^{T}\frac{\lambda_k}{2}b_k^2-\sum_{k=1}^{T}G_k(u_k).
$$
Thus
$$
J(u)\geq \frac{\lambda_{\ell+1}}{2}\|u\|^2-\sum_{k=1}^{T}G_k(u_k).
$$
Further, from (G3), there exists $\sigma>0$ such that
$$
G_k(\xi)\leq \frac{\lambda_{\ell+1}}{2}\xi^2,\quad (\forall\, k\in {\mathbb{Z}}[1,T])
$$
provided $0<|\xi|\leq \sigma$.
Then, taking $ \rho_2\leq{\sigma/N^{1/2}}$, for every 
$u\in\bar{B}({0,\rho_2})\cap X_1$, we have
$$
J(u)\geq \frac{\lambda_{\ell+1}}{2}(\|u\|^2-\|u\|^2)=0.
$$
Therefore, choosing  $\rho\leq \min\{\rho_1,\rho_2\}$, 
if $\inf_{u\in X}J(u)<0=J(0)$ our claim is proved.

 On the other hand, if $ \inf_{u\in X}J(u)= 0$, we argue as above; that is, 
every $u\in X_2$ with $\|u\|_2\leq \rho$ is solution of problem \eqref{W}. 
So, our goal is achieved.
\end{proof}

The following result is a direct consequence of Theorem \ref{Main}. 

\begin{corollary}\label{Main2}
Let $\alpha:{\mathbb{Z}}[1,T]\to \mathbb{R}$ be a nonnegative
 $($not identically zero$)$ function and let $h:\mathbb{R}\to\mathbb{R}$ 
be a locally essentially bounded map.
Assume that there exists an integer $\ell\in {\mathbb{Z}}[1,T-1]$ such that
\begin{itemize}
\item [(G0)] $  \frac{\lambda_{\ell}}{ \sum_{k=1}^{T}\alpha_k}
\leq \lim_{t\to  0}\frac{  h(t)}{t}\leq\frac{\lambda_{\ell+1}}{ \sum_{k=1}^{T}
\alpha_k}$, for all $k\in {\mathbb{Z}}[1,T]$;

\item[(G4)] $ \limsup_{|t|\to  \infty}\frac{ h(t)}{t}
< \frac{\lambda_1}{ \sum_{k=1}^{T}\alpha_k}$.
\end{itemize}
Then the  discrete problem
\begin{equation}  \label{W2} %\tag{$S_{A}^{\alpha,h}$}
\sum_{l=1}^{T}a_{kl}u_l\in \alpha_k[h^-(u_k),h^+(u_k)],
\quad(\forall\,k\in {\mathbb{Z}}[1,T])
\end{equation}
 admits at least two nontrivial solutions.
\end{corollary}

\begin{proof}
It is elementary to observe that from condition (G0) immediately (G2) and (G3) hold.
 We proceed by proving that condition (G4) implies (G1).
Indeed, by (G4), there are constants 
$\epsilon' \in  \big]0,\lambda_1/\big(\sum_{k=1}^{T}\alpha_k\big)\big[$ and
$\sigma>0$ such that
$$
{\frac{h(t)}{t}}< \frac{\lambda_1}{ \sum_{k=1}^{T}\alpha_k}-\epsilon',
$$
for every
$|t|\geq \sigma$. Since $h$ is a locally essentially bounded function,
 we also have
$$
M:=\operatorname{ess\,sup}_{t\in[-\sigma,\sigma]}|h(t)|<+\infty.
$$
Therefore, if $\xi\geq \sigma$, it follows that
$$
\int_0^\xi h(t)dt=\int_0^\sigma h(t)dt+\int_\sigma^\xi h(t)dt
\leq M\sigma+\frac{1}{2}\Big(  {\lambda_1}/\Big({ \sum_{k=1}^{T}\alpha_k}\Big)
-\epsilon'\Big)\xi^2,
$$
 while, for $\xi\leq -\sigma$, one has
$$
\int_0^\xi h(t)dt=-\Big[\int_\xi^{-\sigma} h(t)dt+\int_{-\sigma}^0 h(t)dt\Big]
\leq M\sigma+\frac{1}{2}\Big(  {\lambda_1}/\Big({ \sum_{k=1}^{T}\alpha_k}\Big)
-\epsilon'\Big)\xi^2.
$$
 Consequently,
\begin{equation} \label{Dis}
\int_0^\xi h(t)dt\leq M\sigma+\frac{1}{2}
\Big(  {\lambda_1}/\Big({ \sum_{k=1}^{T}\alpha_k}\Big)-\epsilon'\Big)\xi^2,\quad
 (\forall\, \xi\in\mathbb{R}).
\end{equation}
Hence, by using the above inequality,  we can write
$$
\limsup_{|\xi|\to  \infty}\frac{ \int_0^\xi \alpha_k h(t)dt}{\xi^2}
=\alpha_k \limsup_{|\xi|\to  \infty}\frac{  H(\xi)}{\xi^2}
\leq\frac{1}{2}  \Big(\sum_{k=1}^{T}\alpha_k\Big)
\Big(  {\lambda_1}/\Big({ \sum_{k=1}^{T}\alpha_k}\Big)-\epsilon'\Big)
<\frac{\lambda_1}{2},
$$
for every $k\in {\mathbb{Z}}[1,T]$. So, it is clear that condition (G2) holds.
In conclusion, our claim is verified and the proof is complete.
\end{proof}

\begin{remark}\rm
 Boundary value problems involving fourth-order difference inclusions such as
\begin{equation}  \label{42order} %\tag{$D_{\lambda}^{g}$}
\begin{gathered}
\Delta^4u_{k-2}\in [g^-_k(u_k),g^+_k(u_k)],\quad (\forall\,k\in {\mathbb{Z}}[1,T])\\
{u_{-2}=u_{-1}=u_0=0}\\
u_{T+1}=u_{T+2}=u_{T+3}=0,\\
\end{gathered}
\end{equation}
can also be expressed as problem \eqref{W},
where $A$ is the real symmetric and positive definite matrix of the form
$$
A:=  \begin{pmatrix}
    6 & -4 & 1 & 0 & \dots  & 0 & 0 & 0 & 0 \\
    -4 & 6 & -4 & 1 & \dots  & 0 & 0 &0 & 0 \\
    1 & -4 & 6 & -4 & \dots  & 0 & 0 & 0 & 0 \\
    0 & 1 & -4 & 6 & \dots  & 0 & 0 & 0 & 0 \\
     &  &  &  & \ddots &  &  &  &  \\
    0 & 0 & 0 & 0 & \dots  & 6 & -4 & 1 & 0 \\
    0 & 0 & 0 & 0 & \dots  & -4 & 6 & -4 & 1 \\
    0 & 0 & 0 & 0 & \dots  & 1 & -4 & 6 & -4 \\
    0 & 0 & 0 & 0 & \dots  & 0 & 1 & -4 & 6 \\
  \end{pmatrix}
$$
in $\mathfrak{X}_{T}$.
 Then, it is easily seen that Theorem \ref{Introduction} in the introduction 
is a direct consequence of Corollary \ref{Main2}.
\end{remark}

\section{Partial algebraic inclusions}

Nonlinear inclusions of the
form \eqref{W} arise in many applications such as boundary value problems
involving partial difference equations.
For instance, we just point out that our results can be applied to the 
following problem 
%namely $(E^f)$, given as follows
\begin{equation} \label{Ef}
\begin{aligned}
&4u(i,j)-u(i+1,j)-u(i-1,j)-u(i,j+1)-u(i,j-1)\\
&\in[f^-_{(i,j)}(u(i,j)),f^+_{(i,j)}(u(i,j))]
\end{aligned}
\end{equation}
for every $(i,j)\in {\mathbb{Z}}[1,m]\times{\mathbb{Z}}[1,n]$,
with boundary conditions
\begin{gather*}
u(i,0)=u(i,n+1)=0,\quad (\forall\, i\in {\mathbb{Z}}[1,m]),\\
u(0,j)=u(m+1,j)=0,\quad (\forall\, j\in {\mathbb{Z}}[1,n])
\end{gather*}
where every $f_{(i,j)}:\mathbb{R}\to  \mathbb{R}$ denotes a locally
essentially bounded function.

Let $z:{\mathbb{Z}}[1,m]\times{\mathbb{Z}}[1,n]\to  {\mathbb{Z}}[1,mn]$ 
be the bijection defined by $z(i,j):=i+m(j-1)$, for every 
$(i,j)\in {\mathbb{Z}}[1,m]\times{\mathbb{Z}}[1,n]$.
Let us denote $w_k:=u({z^{-1}(k)})$ and $g_k(w_k):=f_{z^{-1}(k)}(w_k)$, 
for every $k\in {\mathbb{Z}}[1,mn]$.

With the above notation, problem \eqref{Ef} can be written as a nonlinear 
algebraic inclusion of the form
\begin{equation} \label{W6} %\tag{$S_{B}^{g}$} 
\sum_{l=1}^{T}b_{kl}w_l\in[g^-_k(w_k),g^+_k(w_k)],\quad
(\forall\,k\in {\mathbb{Z}}[1,mn]),
\end{equation}
where
$$
B:=(b_{ij})=
  \begin{pmatrix}
    L & -I_m & 0 & 0 & \dots  & 0 & 0 & 0 & 0 \\
    -I_m & L & -I_m & 0 & \dots  & 0 & 0 &0 & 0 \\
    0 & -I_m & L & -I_m & \dots  & 0 & 0 & 0 & 0 \\
    0 & 0 & -I_m & L & \dots  & 0 & 0 & 0 & 0 \\
     &  &  &  & \ddots &  &  &  &  \\
    0 & 0 & 0 & 0 & \dots  & L & -I_m & 0 & 0 \\
    0 & 0 & 0 & 0 & \dots  & -I_m & L & -I_m & 0 \\
    0 & 0 & 0 & 0 & \dots  & 0 & -I_m & L & -I_m \\
    0 & 0 & 0 & 0 & \dots  & 0 & 0 & -I_m & L \\
  \end{pmatrix}
$$
in $\mathfrak{X}_{mn}$,
in which $L$ is the $m\times m$ matrix 
$$
L:=\begin{pmatrix}
    4 & -1 & 0 & 0 & \dots  & 0 & 0 & 0 & 0 \\
    -1 & 4 & -1 & 0 & \dots  & 0 & 0 &0 & 0 \\
    0 & -1 & 4 & -1 & \dots  & 0 & 0 & 0 & 0 \\
    0 & 0 & -1 & 4 & \dots  & 0 & 0 & 0 & 0 \\
     &  &  &  & \ddots &  &  &  &  \\
    0 & 0 & 0 & 0 & \dots  & 4 & -1 & 0 & 0 \\
    0 & 0 & 0 & 0 & \dots  & -1 & 4 & -1 & 0 \\
    0 & 0 & 0 & 0 & \dots  & 0 & -1 & 4 & -1 \\
    0 & 0 & 0 & 0 & \dots  & 0 & 0 & -1 & 4 \\
  \end{pmatrix}
$$
 and $I_m$ is the $m\times m$ identity matrix.
 
Finally for completeness, we observe that the existence of
multiple solutions for the nonlinear discrete problems can be 
used in the study of numerical methods applied to some mathematical models;
see for instance the recent article \cite{PFD}.

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