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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 148, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/148\hfil Anisotropic discrete BVPs]
{Existence results for  anisotropic discrete boundary value problems}

\author[M. Avci \hfil EJDE-2016/148\hfilneg]
{Mustafa Avci}

\address{Mustafa Avci \newline
 Faculty of Economics and Administrative Sciences,
Batman University, Turkey}
\email{avcixmustafa@gmail.com}

\thanks{Submitted March 23, 2016. Published June 16, 2016.}
\subjclass[2010]{47A75, 35B38, 35P30, 34L05, 34L30}
\keywords{Anisotropic discrete boundary value problems; 
\hfill\break\indent multiple solutions;
variational methods; critical point theory}

\begin{abstract}
 In this article, we prove the existence of nontrivial weak solutions for a
 class of discrete boundary value problems. The main tools used here
 are the variational principle and critical point theory.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks



\section{Introduction and Preliminaries}

In this article, we are interested in the existence of solutions for the
 discrete boundary value problem
\begin{equation}
\begin{gathered}
-\Delta ( |\Delta u(k-1)|^{p(k-1)-2}\Delta u(k-1)) =\lambda
f(k,u(k)),\quad k\in \mathbb{Z}[1,T], \\
u(0)=u(T+1)=0,
\end{gathered} \label{e1.1}
\end{equation}
where $T\geq 2$ is a positive integer; $\mathbb{Z}[a,b]$ denotes the
discrete interval $\{a,a+1,\dots,b\}$ with $a$ and $b$ are integers such that
$a<b$; $\Delta u(k)=u(k+1)-u(k)$ is the forward difference operator;
$\lambda$ is a positive parameter;
$f:\mathbb{Z}[1,T]\times \mathbb{R}\to \mathbb{R}$ is a continuous function
with respect to  $t\in\mathbb{R}$, and $k\in \mathbb{Z}[1,T]$.
For the function $p:\mathbb{Z}[0,T]\to [ 2,\infty )$ denote
\[
p^{-}:=\min_{k\in \mathbb{Z}[0,T]}p(k)\leq p(k)\leq \max_{k\in \mathbb{Z}
[0,T]}p(k)=:p^{+}<\infty .
\]
In the previous decades, the nonlinear difference equations have been
intensively used for the mathematical modelling of various problems in
different disciplines of science, such as computer science, mechanical
engineering, control systems, artificial or biological neural networks and
economics. This mades nonlinear difference equations very attractive
to many authors, and hence, many paper have been devoted to the relative
field by using a various methods such as fixed points theorems, topological
methods and variational methods.
For the recent progress in discrete problems, we refer the readers to the
interesting book by Agarwal \cite{Agarwal} and the papers \cite{CaIaTe,MiRaTe}.

In \cite{AgPeOr,AvcPan,BiRe,BoCa,CaIaTe,CaGi,LiWaZhZh,YaZh}, the authors used
different methods to study the existence and multiplicity of solutions for
the discrete boundary value problem of the  type
\begin{equation}
\begin{gathered}
-\Delta \big( \phi _{p}(\Delta u(k-1))\big) =f(k,u(k)),\quad k\in \mathbb{
Z}[1,T], \\
u(0)=u(T+1)=0,
\end{gathered}\label{e1.2}
\end{equation}
where $\phi _{p}(s)=|s|^{p-2}s$, $1<p<+\infty $. In \cite{MiRaTe},
Mih\u{a}ilescu et al. studied the eigenvalue problem for the
anisotropic discrete boundary-value problem
\begin{equation}
\begin{gathered}
-\Delta \big( \phi _{p(k-1)}(\Delta u(k-1))\big)
=\lambda |\Delta u(k-1)|^{q(k-1)-2}\Delta u(k-1),\quad k\in \mathbb{Z}[1,T], \\
u(0)=u(T+1)=0,
\end{gathered} \label{e1.3}
\end{equation}
where $\phi _{p(k-1)}(s)=|s|^{p(k-1)-2}s$,
$p:\mathbb{Z}[0,T]\to [ 2,+\infty )$,
$q:\mathbb{Z}[1,T]\to [ 2,+\infty )$ and
$\lambda $ is a positive parameter. For the recent papers involving
anisotropic discrete boundary value problems, we refer
to recent works \cite{BeJeSe,GaGl,GaWi, KoOu,MaYuOg} and references therein.
Motivated by the papers mentioned above, we study problem \eqref{e1.1}
and obtain the existence of nontrivial weak solutions
by employing variational principle and critical point theory argued
in \cite{Bonanno}.

Let us define the function space
\[
W=\{ u:\mathbb{Z}[0,T+1]\to \mathbb{R}\text{ such that }
u(0)=u(T+1)=0\} .
\]
Then, $W$ is a $T$-dimensional Hilbert space with the inner product
\[
(u,v)=\sum_{k=1}^{T+1}\Delta u(k-1)\Delta v(k-1),\quad \forall u,v\in W,
\]
while the corresponding norm is given by
\[
\| u\| _{W}=\Big( \sum_{k=1}^{T+1}|\Delta u(k-1)|^{2}\Big) ^{1/2}.
\]
We can also define the following norm on $W$ since $W$ is
finite-dimensional,
\[
|u|_{m}=\Big( \sum_{k=1}^{T}|u(k)|^{m}\Big) ^{1/m},\quad \forall
u\in W,\; m\geq 2.
\]
Now, we recall some auxiliary results which we use through the paper.

\begin{proposition}[\cite{GaGl,MiRaTe}] \label{pro1.1}
(i) Let $u\in W$ and $\| u\| _{W}>1$. Then
\[
\sum_{k=1}^{T+1}\frac{|\Delta u(k-1)|^{p(k-1)}}{p(k-1)}\geq \frac{1}{p^{+}(
\sqrt{T})^{p^{-}-2}}\| u\| _{W}^{p^{-}}-T.
\]
(ii) Let $u\in W$ and $\| u\| _{W}<1$. Then
\[
\sum_{k=1}^{T+1}\frac{|\Delta u(k-1)|^{p(k-1)}}{p(k-1)}\geq \frac{1}{p^{+}(
\sqrt{T})^{2-p^{+}}}\| u\| _{W}^{p^{+}}.
\]
(iii) For any $m\geq 2$, there exist positive constants $c_{m}$ such
that
\[
\sum_{k=1}^{T}|u(k)|^{m}\leq c_{m}\sum_{k=1}^{T+1}|\Delta u(k-1)|^{m},\quad
\forall u\in W.
\]
(iv)  For any $m\geq 2$, we have
\[
( T+1) ^{\frac{2-m}{2}}\| u\| _{W}^{m}\leq
\sum_{k=1}^{T+1}|\Delta u(k-1)|^{m}\leq ( T+1) \| u\|
_{W}^{m},\quad  \forall u\in W.
\]
(v)  For any $m\geq 2$, we have
\[
2^{m}\sum_{k=1}^{T}|u(k)|^{m}\geq \sum_{k=1}^{T+1}|\Delta u(k-1)|^{m},\quad
\forall u\in W.
\]
\end{proposition}

The key arguments in our paper are the following results given in
\cite{Bonanno}.

\begin{proposition}\label{pro1.2}
Let $X$ be a real Banach space, $\Phi ,\Psi:X\to \mathbb{R} $
be two continuously G\^{a}teaux differentiable functionals such
that $\inf_{x\in X}\Phi (x)=\Phi (0)=\Psi (0)=0$. Assume that there
exist $r>0$ and $\overline{x}\in X$, with $0<\Phi (\overline{x})<r$,
such that:
\begin{itemize}
\item[(1)] $\frac{1}{r}\sup_{\Phi (x)\leq r}\Psi (x)\leq \frac{\Psi (
\overline{x})}{\Phi (\overline{x})}$,

\item[(2)] for each $\lambda \in \Lambda _{r}:=\big( \frac{\Phi (
\overline{x})}{\Psi (\overline{x})},\frac{r}{\sup_{\Phi (x)\leq r}\Psi (x)}
\big) $, the functional $I_{\lambda }:=\Phi -\lambda \Psi$  satisfies the
$(P.S.)^{[r]}$ condition.

\end{itemize}
Then, for each $\lambda \in \Lambda _{r}$, there is
$x_{0,\lambda }\in \Phi ^{-1}( ( 0,r) ) $ such
that $I_{\lambda }'( x_{0,\lambda }) \equiv \vartheta
_{X^{\ast }}$ and $I_{\lambda }( x_{0,\lambda }) \leq
I_{\lambda }( x) $ for all $x\in \Phi ^{-1}(( 0,r) ) $.
\end{proposition}

\begin{proposition} \label{pro1.3}
Let $X$ be a real Banach space, $\Phi ,\Psi:X\to\mathbb{R}$
 be two continuously G\^{a}teaux differentiable functionals such
that $\Phi $ is bounded from below and
$\Phi ( 0) =\Psi ( 0) =0$. Fix $r>0$  and assume that for
each
\[
\lambda \in \Big( 0,\frac{r}{\sup_{u\in \Phi ^{-1}( ( -\infty
,r) ) }\Psi ( u) }\Big) ,
\]
the functional  $I_{\lambda }:=\Phi -\lambda \Psi$  satisfies the
$(P.S.)$ condition and it is unbounded from below. Then for each
\[
\lambda \in \Big( 0,\frac{r}{\sup_{u\in \Phi ^{-1}( ( -\infty
,r) ) }\Psi ( u) }\Big) ,
\]
the functional $I_{\lambda }$ admits two distinct critical
points.
\end{proposition}

\begin{proposition} \label{pro1.4} Let $X$ be a reflexive real Banach space,
$\Phi :X\to\mathbb{R}$ be a coercive, continuously G\^{a}teaux differentiable and
sequentially weakly lower semi-continuous functional whose G\^{a}teaux
derivative admits a continuous inverse on $X^{\ast }$;
$\Psi :X\to\mathbb{R}$ be a continuously G\^{a}teaux differentiable
functional whose G\^{a}teaux derivative is compact such that
$\inf_{x\in X}\Phi (x)=\Phi(0)=\Psi (0)=0$.  Assume that there exists
$r>0$ and $\overline{x}\in X$, with $r<\Phi (\overline{x})$ such that:
\begin{itemize}
\item[(1)] $\frac{1}{r}\sup_{\Phi (x)\leq r}\Psi (x)\leq \frac{\Psi (
\overline{x})}{\Phi (\overline{x})}$,

\item[(2)] for each 
\[
\lambda \in \Lambda _{r}:=\Big( \frac{\Phi (
\overline{x})}{\Psi (\overline{x})},\frac{r}{\sup_{\Phi (x)\leq r}\Psi (x)}
\Big) ,
\]
 the functional  $I_{\lambda }:=\Phi -\lambda \Psi$  is coercive.

\end{itemize}
Then, for each $\lambda \in \Lambda _{r}$, the functional
$I_{\lambda }$ has at least three distinct critical points.
\end{proposition}

Let us proceed with setting  problem \eqref{e1.1} in the
variational structure. To this end, let us define the functionals $\Phi ,\Psi
:W\to \mathbb{R}$ as follows:
\begin{gather*}
\Phi (u)=\sum_{k=1}^{T+1}\frac{|\Delta u(k-1)|^{p(k-1)}}{p(k-1)}, \\
\Psi ( u) =\sum_{k=1}^{T}F(k,u(k)),
\end{gather*}
where $F(k,s)=\int_0^{s}f( k,t) dt$,
$( k,s) \in \mathbb{Z}[1,T]\times\mathbb{R}$.

The functionals $\Phi $ and $\Psi $ are well-defined and continuously
G\^{a}teaux differentiable where their derivatives are
\begin{gather*}
\Phi '(u)\varphi =\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p(k-1)-2}\Delta
u(k-1)\Delta \varphi (k-1),
\\
\Psi '(u)\varphi =\sum_{k=1}^{T}f(k,u(k))\varphi (k),
\end{gather*}
 for all $u,\varphi \in W$.

Then the functional $I_{\lambda }:W\to\mathbb{R}$ corresponding to
problem \eqref{e1.1} is
\[
I_{\lambda }( u) :=\Phi ( u) -\lambda \Psi (u).
\]
The functional $I_{\lambda }$ is also well defined on $W$
and $I_{\lambda}\in C^{1}( W, \mathbb{R})$ with the derivative
\[
I_{\lambda }'( u) \varphi =\Phi '(u)\varphi
-\lambda \Psi '( u) \varphi ,
\]
for all $u,\varphi \in W$.

We want to remark that since problem \eqref{e1.1} is defined in a
finite-dimensional Hilbert space $W$, it is not difficult to verify that the
functionals $\Phi $, $\Psi $ and $I_{\lambda }$ satisfy the regularity
assumptions mentioned above (see, e.g., \cite{Jiang}).

\begin{definition} \label{def.1.5} \rm
We say that $u\in W$ is a weak solution of problem \eqref{e1.1} if
\begin{equation}
\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p(k-1)-2}\Delta u(k-1)\Delta \varphi
(k-1)-\lambda \sum_{k=1}^{T}f(k,u(k))\varphi (k)=0  \label{e1.4}
\end{equation}
for all $\varphi \in W$, where \eqref{e1.4} is called the weak form of
problem \eqref{e1.1}.
\end{definition}

From the above definition it is obvious that the weak solutions of problem
\eqref{e1.1} are in fact the critical points of $I_{\lambda }$.

We also use the following helpful notation:
\begin{equation}
\begin{gathered}
\beta ^{p_{\ast }}=\begin{cases}
\beta ^{p^{+}} & \text{if } \beta >1 \\
\beta ^{p^{-}} & \text{if } 0<\beta <1,
\end{cases} \quad
\eta ^{1/p_{\ast }}=\begin{cases}
\eta ^{1/p^{-}} & \text{if }\eta >1 \\
\eta ^{1/p^{+}} & \text{if }0<\eta <1,
\end{cases}
\\
\delta ^{( q/p) _{\ast }}=\begin{cases}
\delta ^{q^{+}/p^{-}} & \text{if }\delta >1 \\
\delta ^{q^{-}/p^{+}} & \text{if }0<\delta <1\,.
\end{cases}
\end{gathered} \label{e1.5}
\end{equation}

\section{Existence of one solution}

We sue the following assumptions:
\begin{itemize}
\item[(A1)] There exist $C>0$ and a function $q:\mathbb{Z}[1,T]\to
[ 2,+\infty )$ such that for all $(k,t)\in \mathbb{Z}[1,T]\times
\mathbb{R}$,
\[
|f(k,t)|\leq C\big( 1+|t|^{q(k)-1}\big) .
\]

\item[(A2)] There exist $r,a,b,l>0$ with
\[
b<( \frac{p^{-}}{2})
^{1/p^{+}}\frac{a}{( T+1) ^{\frac{^{( p^{+}-2) }}{
2p^{-}}}( p^{+}) ^{\frac{1}{p^{-}}}}
\]
 such that
\[
\frac{l}{r}( T+( T+1) ^{( p^{+}-2)
q^{+}/2p^{-}}( p^{+}) ^{q^{+}/p^{-}}r^{( q/p) _{\ast
}}) <b_{p}\sum_{k=1}^{T}F(k,b),
\]
where
\[
b_{p}=\Big( \frac{b^{p( 0) }}{^{p( 0) }}+\frac{
b^{p( T) }}{^{p( T) }}\Big) ^{-1}.
\]
\end{itemize}

\begin{theorem} \label{the2.1}
Assume {\rm (A1)} and {\rm (A2)} are satisfied.
Then for each
\[
\lambda \in \Lambda _{r,b}:=\Big( \frac{1}{
b_{p}\sum_{k=1}^{T}F(k,b)},\frac{r}{l( T+( T+1) ^{(
p^{+}-2) q^{+}/2p^{-}}( p^{+}) ^{q^{+}/p^{-}}r^{(
q/p) _{\ast }}) }\Big) ,
\]
problem \eqref{e1.1}
 admits at least one nontrivial weak solution.
\end{theorem}

\begin{proof}
We will apply Proposition \ref{pro1.2}. We know that $\Phi $ and $\Psi $ are
well-defined and continuously G\^{a}teaux differentiable. Moreover, from the
definitions of $\Phi $ and $\Psi $ we have
\[
\inf_{u\in W}\Phi (u)=\Phi ( 0) =\Psi ( 0) =0.
\]

Let us define the function $\overline{u}:\mathbb{Z}[0,T+1]\to\mathbb{R}$
belonging to $W$ by the formula
\[
\overline{u}( k) =\begin{cases}
b & \text{if }k\in \mathbb{Z}[1,T], \\
0 & \text{if }k=0,k=T+1.
\end{cases}
\]
Then, we deduce that
\[
\Phi ( \overline{u}) =\frac{b^{p( 0) }}{^{p(
0) }}+\frac{b^{p( T) }}{^{p( T) }}
\]
which implies
\[
\Phi ( \overline{u}) \leq \frac{2}{p^{-}}b^{p_{\ast}}.
\]
Moreover, we have
\[
\frac{\Psi ( \overline{u}) }{\Phi ( \overline{u}) }=
\frac{\sum_{k=1}^{T}F(k,b)}{\frac{b^{p( 0) }}{^{p( 0) }
}+\frac{b^{p( T) }}{^{p( T) }}}.
\]
For each $u\in \Phi ^{-1}( ( -\infty ,r) ) $, from
Proposition \ref{pro1.1}(iv) and \ref{e1.5}, one has
\begin{gather*}
\frac{1}{p^{+}}( T+1) ^{( 2-p^{+}) /2}\| u\|
_{W}^{p_{\ast }}\leq \frac{1}{p^{+}}\sum_{k=1}^{T+1}|\Delta
u(k-1)|^{p(k-1)}\leq \Phi (u)\leq r
\\
\| u\| _{W}\leq ( ( T+1) ^{(
p^{+}-2) /2}p^{+}r) ^{1/p_{\ast }}\leq ( T+1)
^{( p^{+}-2) /2p^{-}}( p^{+}) ^{1/p^{-}}r^{1/p_{\ast}}:=a
\end{gather*}
Then
\begin{equation}
r=\frac{a^{p_{\ast }}}{( T+1) ^{\frac{^{( p^{+}-2)
p_{\ast }}}{2p^{-}}}( p^{+}) ^{\frac{p_{\ast }}{p^{-}}}}.
\label{e2.1}
\end{equation}
Since
\[
b<\big( \frac{p^{-}}{2}\big) ^{1/p^{+}}\frac{a}{( T+1)
^{\frac{^{( p^{+}-2) }}{2p^{-}}}( p^{+}) ^{\frac{1}{
p^{-}}}},
\]
we obtain $\Phi ( \overline{u}) <r$.
Moreover, from condition (A1), there exists a constant $l>0$ such that
 $|F(k,t)|\leq l( 1+|t|^{q(k)}) $. Then it follows
\begin{gather*}
\Psi ( u) \leq \sum_{k=1}^{T}l( 1+| u(k) | ^{q( k) })
\leq l( T+\| u\| _{W}^{q_{\ast }}) ,
\\
\| u\| _{W}^{q_{\ast }}\leq ( T+1) ^{(
p^{+}-2) q^{+}/2p^{-}}( p^{+}) ^{q^{+}/p^{-}}r^{(
q/p) _{\ast }},
\end{gather*}
and hence
\begin{equation}
\frac{1}{r}\sup_{\Phi ( u) \leq r}\Psi ( u) \leq \frac{
l}{r}\Big( T+( T+1) ^{( p^{+}-2) q^{+}/2p^{-}}(
p^{+}) ^{q^{+}/p^{-}}r^{( q/p) _{\ast }}\Big) .
\label{e2.2}
\end{equation}
Taking into account the condition (A2), we have
\[
\frac{1}{r}\sup_{\Phi ( u) \leq r}\Psi ( u) \leq \frac{
\Psi ( \overline{u}) }{\Phi ( \overline{u}) }.
\]
In conclusion, Proposition \ref{pro1.2}(1) is verified.

For Proposition \ref{pro1.2}(2), as mentioned before, $\Phi $ and 
$\Psi $ are well-defined and continuously G\^{a}teaux differentiable. 
Further, from (A1), $\Psi $
has a compact derivative. This ensures that the functional $I_{\lambda }$
satisfies the $(P.S.)^{[r]}$ condition for each $r>0$. Hence Proposition
 \ref{pro1.2}(2) is verified as well.

Consequently, by Proposition \ref{pro1.2}, for each 
$\lambda \in \Lambda _{r,b}$, the functional $I_{\lambda }$ admits at 
least one critical point which corresponds to the nontrivial weak solution 
of problem \eqref{e1.1}.
\end{proof}

\begin{example} \label{examp1}  \rm
As an  application of Theorem \ref{the2.1}, we consider the
following: Let $T=2$, $p( k-1) =2( k+1) $, $q(k) =k+1$, $b=1$, $a=7$ and 
$f( k,u) =| u(k) | ^{k}$. Then, $p^{-}=p( 0) =4$, 
$p^{+}=p( T) =6$, $q^{-}=2$, $q^{+}=3$, 
$F( k,u) =\frac{1}{k+1}| u( k) | ^{k+1}$ and $l=\frac{1}{k+1}$, say 
$l=1/3$. Then, the all the assumptions requested in Theorem
\ref{the2.1} hold. Finally, by simple computations, it results that 
for each $\lambda \in \Lambda _{r,b}\subseteq (1/2,\lambda_a) $
problem \eqref{e1.1} admits at least
one nontrivial weak solution, where the real constant $\lambda _a$
depends on $a$ and satisfies $\lambda _a\geq 31/50$.
\end{example}

\section{Existence of two solutions}

For the next theorem we use the assumption
\begin{itemize}
\item[(A3)] There exist positive real numbers $\theta$ and $t_0$ such that
$\theta >p^{+}$ and
\[
0<\theta F( k,t) \leq f( k,t) t\quad \forall
( k,t) \in \mathbb{Z}[1,T]\times\mathbb{R},\; | t| \geq t_0\,.
\]
\end{itemize}


\begin{theorem}\label{the3.1}
Assume that {\rm (A1)} and {\rm (A3)} hold.
 Then for each 
\[
\lambda \in \Lambda _{r}:=\Big( 0,\frac{r}{l(
T+( T+1) ^{( p^{+}-2) q^{+}/2p^{-}}( p^{+})
^{q^{+}/p^{-}}r^{( q/p) _{\ast }}) }\Big),
\]
problem \eqref{e1.1} admits at least two distinct weak
solutions.
\end{theorem}

\begin{proof}
We will apply Proposition \ref{pro1.3}. It is obvious that 
$\Phi (0) =\Psi ( 0) =0$. Moreover, $\Phi $ is bounded from below.
Indeed, for $\| u\| _{W}<1$ and by Proposition \ref{pro1.1}(ii), it reads
\[
\Phi (u) =\sum_{k=1}^{T+1}\frac{1}{p(k-1)}|\Delta u(k-1)|^{p(k-1)} 
\geq \frac{1}{p^{+}(\sqrt{T})^{2-p^{+}}}\| u\| _{W}^{p^{+}}.
\]
Let us show that $I_{\lambda }$ is unbounded from below and satisfies 
the $(P.S.)$ condition. From condition (A3), there exists a constant 
$c>0$ such that $F(k,t)\geq c|t|^{\theta }$ for any 
$(k,t)\in \mathbb{Z}[1,T]\times \mathbb{R}$, $|t| \geq t_0$.
Let $\| u\| _{W}>1$. Then, using 
Proposition \ref{pro1.1}(iv)-(v), it reads
\begin{align*}
I_{\lambda }(u) 
&\leq \frac{1}{p^{-}}\sum_{k=1}^{T+1}|\Delta
u(k-1)|^{p(k-1)}-\lambda \sum_{k=1}^{T}F(k,u(k)) \\
&\leq \frac{( T+1) }{p^{-}}\| u\| _{W}^{p^{+}}-\lambda
c\sum_{k=1}^{T}| u( k) | ^{\theta } \\
&\leq \frac{( T+1) }{p^{-}}\| u\| _{W}^{p^{+}}-\lambda
c2^{-\theta }\sum_{k=1}^{T+1}|\Delta u(k-1)|^{\theta } \\
&\leq \frac{( T+1) }{p^{-}}\| u\| _{W}^{p^{+}}-\lambda
c2^{-\theta }( T+1) ^{( 2-\theta ) /2}\| u\|
_{W}^{\theta },
\end{align*}
from which we get
\[
\lim_{\| u\| _{W}\to \infty }I_{\lambda }( u)=-\infty.
\]
Therefore, $I_{\lambda }$ is unbounded from below and anti-coercive. Additionally,
since the space $W$ is finite-dimensional, $(P.S.)$ condition
follows immediately. Consequently, all assumptions of Proposition
\ref{pro1.3} are verified. Therefore, for each $\lambda \in \Lambda _{r}$, the
functional $I_{\lambda }$ admits two distinct critical points that are weak
solutions of problem $(1.1)$.
\end{proof}

We want to remark that from condition (A1), there exists a constant 
$C_1>0$ such that\ $|F(k,t)|\leq C_1( 1+|t|^{q(k)}) $, and from condition (A3),
 there exists a constant $C_2>0$ such that\ $F(k,t)\geq C_2|t|^{\theta }$
for any $(k,t)\in \mathbb{Z}[1,T]\times \mathbb{R}$. So, we conclude that 
 $C_2|t|^{\theta}\leq C_1( 1+|t|^{q(k)}) $ which means that
$q( k) \geq \theta $ for all $k\in \mathbb{Z}[1,T]$. Therefore, we have 
$p^{+}<\theta \leq q^{-}$ as a natural condition raised from (A1) and (A3).

\begin{example} \label{examp2} \rm
As an application of Theorem \ref{the3.1}, we consider the
function 
\[
f( k,t) =\begin{cases}
m+nq( k) t^{q( k) -1} & t\geq 0, \\
m-nq( k) ( -t) ^{q( k) -1} & t<0,
\end{cases}
\]
for each $(k,t)\in \mathbb{Z}[1,T]\times \mathbb{R}$ where $m,n$ are some
positive constants. We also assume that
\[
t_0>\max \Big\{ \Big( \frac{
m( \theta -1) }{n( q^{-}-\theta ) }\Big) ^{\frac{1}{
q( k) -1}},\Big( \frac{m}{n}\Big) ^{\frac{1}{q( k)
-1}}\Big\}
\]
 such that $ q^{-} > \theta $. Then, condition (A1) is easily verified.
Let us proceed for condition (A3). From the above definition of $f$, we get
 $ F(k,t)=mt+n| t| ^{q( k) }$. Since we have
$t_0^{q( k) -1}>\frac{m}{n}$, for all $k\in \mathbb{Z}[1,T]$ and
$| t| \geq t_0$ there holds
\[
F(k,t)\geq | t| \big( -m+n|t|^{q(k)-1}\big)
\geq t_0( -m+nt_0^{q(k)-1}) >0.
\]

Moreover, for all $k\in \mathbb{Z}[1,T]$ and $t<0$, we have
\begin{align*}
tf( k,t) -\theta F(k,t) 
&= m( 1-\theta ) t+n(q( k) -\theta ) | t| ^{q( k) }\\
&= m( \theta -1) | t| +n( q(k) -\theta ) | t| ^{q( k) }>0.
\end{align*}
Finally, thanks to the assumption 
$t_0^{q( k) -1}>\frac{m(\theta -1) }{n( q^{-}-\theta ) }$, for all 
$k\in \mathbb{Z}[1,T]$ and $t\geq t_0$, we obtain
\begin{align*}
tf( k,t) -\theta F(k,t) 
&=t( n( q( k)-\theta ) | t| ^{q( k) -1}-m(\theta -1) ) \\
&\geq t_0( n( q^{-}-\theta ) r^{q( k)-1}-m( \theta -1) ) >0.
\end{align*}
Therefore condition (A3) holds as well.
\end{example}

\section{Existence of three solutions}

For the next theorem we use the assumption
\begin{itemize}
\item[(A4)] There exist $C_{d},d>0$ with
\[
d>( \frac{p^{+}}{2})
^{1/p^{+}}\frac{a}{( T+1) ^{\frac{^{( p^{+}-2) }}{
2p^{-}}}( p^{+}) ^{\frac{1}{p^{-}}}}
\]
 such that
\[
\frac{l}{r}( T+( T+1) ^{( p^{+}-2)
q^{+}/2p^{-}}( p^{+}) ^{q^{+}/p^{-}}r^{( q/p) _{\ast
}}) <C_{d}d_{p}\sum_{k=1}^{T}F(k,d),
\]
where $d_{p}=\big( \frac{d^{p( 0) }}{^{p( 0) }}+\frac{
d^{p( T) }}{^{p( T) }}\big) ^{-1}$.
\end{itemize}

\begin{theorem} \label{the4.1}
Assume  {\rm (A1), (A4)} and $q^{+}<p^{-}$.
Then for each 
\[
\lambda \in \Lambda _{r,d}:=\Big( \frac{1}{
C_{d}d_{p}\sum_{k=1}^{T}F(k,d)},\frac{r}{l( T+( T+1)
^{( p^{+}-2) q^{+}/2p^{-}}( p^{+})
^{q^{+}/p^{-}}r^{( q/p) _{\ast }}) }\Big),
\]
problem \eqref{e1.1}  admits at least three distinct weak
solutions.
\end{theorem}

\begin{proof}
We will apply Proposition \ref{pro1.4}. We know that, $\Phi $ and $\Psi $
are well-defined and continuously G\^{a}teaux differentiable, and 
$\inf_{u\in W}\Phi (u)=\Phi ( 0) =\Psi ( 0) =0$. The
compactness of derivative of $\Psi $ follows from the growth condition (A1).
Since $\Phi $ is of class $C^{1}$ on the finite-dimensional Hilbert space
 $W$,  to prove that $\Phi $ is weakly lower semicontinuous, it is
sufficient to show the coercivity of $\Phi $ (see \cite{Galewski}). 
Indeed, let $u\in W$ such that $\| u\| _{W}\to +\infty $. Then, without loss of
generality, we can assume that $\| u\| _{W}>1$. From the definition of
the functional $\Phi $ and Proposition \ref{pro1.1}(i), we deduce that
\[
\Phi (u) =\sum_{k=1}^{T+1}\frac{1}{p(k-1)}|\Delta u(k-1)|^{p(k-1)} 
\geq \frac{1}{p^{+}(\sqrt{T})^{p^{-}-2}}\| u\| _{W}^{p^{-}}-T.
\]
So, $\Phi (u)\to +\infty $ as $\| u\| _{W}\to +\infty $
which means that $\Phi $ is coercive. We continue to show the existence of the
inverse function $(\Phi ')^{-1}:W^{\ast }\to W$. At first, we show the 
strict monotonicity of $\Phi '$. For the case $u_1\neq u_2\in W$, we have
\begin{align*}
&( \Phi '(u_1)-\Phi '(u_2)) (
u_1-u_2) \\
&\geq \sum_{k=1}^{T+1}( |\Delta u_1(k-1)|^{p(k-1)-2}\Delta
u_1(k-1)-|\Delta u_2(k-1)|^{p(k-1)-2}\Delta u_2(k-1)) \\
&\quad \times (\Delta u_1(k-1)-\Delta u_2(k-1))
\end{align*}
By the well-known inequality, for any $\zeta ,\xi \in \mathbb{R}^{N}$,
\[
( | \zeta | ^{r-2}\zeta -| \xi |
^{r-2}\xi ) ( \zeta -\xi ) \geq C_{r}| \zeta -\xi
| ^{r},\quad r\geq 2,\; C_{r}>0,
\]
we obtain
\[
( \Phi '(u_1)-\Phi '(u_2)) (u_1-u_2)
\geq c_{3}\sum_{k=1}^{T+1}|\Delta u_1(k-1)-\Delta
u_2( k-1) |^{p(k-1)-2}
>0,
\]
where $c_{3}$ is a positive constant depends only on $p$.
Therefore $\Phi'$ is strictly monotone, which ensures that $\Phi '$ is an
injection. Moreover, by Proposition \ref{pro1.1}, we have
\[
\Phi '(u)u =\sum_{k=1}^{T+1}|\Delta u(k-1)|^{p(k-1)} 
\geq c_{4}\min \big\{\| u\| _{W}^{p^{-}},\| u\|_{W}^{p^{+}}\big\}-c_{5},
\]
where $c_{4},c_{5}$ are positive constants and $u\in W$. So,
$\Phi '(u)\to +\infty $ as $\| u\| _{W}\to +\infty $.

From the above information, and Minty-Browder theorem (see \cite{Zeidler}),
we obtain that $\Phi '$ is a surjection. As a consequence, 
$\Phi'$ has an inverse mapping $(\Phi ')^{-1}:W^{\ast}\to W$. 
We now show that $( \Phi ') ^{-1}$ is
continuous. To this end, let $(u_n^{\ast }),u^{\ast }\in W^{\ast }$ with
$u_n^{\ast }\to u^{\ast }$, and let
$( \Phi ')^{-1}(u_n^{\ast })=(u_n),( \Phi ') ^{-1}(u^{\ast})=u $.
Then, $\Phi (u_n)=u_n^{\ast }$ and $\Phi (u)=u^{\ast }$, which
means that $( u_n) $ is bounded in $W$. Hence there exists
$u_0\in W $ and a subsequence, again denoted by $( u_n) $,
such that $u_n\rightharpoonup u_0$ in $W$, and therefore
$u_n\to u_0$ in $W$. Since the limit is unique, it follows that 
$u_n\to u$ in $W$. Therefore $( \Phi ') ^{-1}$ is continuous.

Next we verify Proposition \ref{pro1.4}(1). 
To do this, let us define the function 
$\overline{v}:\mathbb{Z}[0,T+1]\to\mathbb{R}$ belonging to $W$ by the formula
\[
\overline{v}( k) =\begin{cases}
d & \text{if }k\in \mathbb{Z}[1,T], \\
0 & \text{if }k=0,k=T+1,
\end{cases}
\]
Then we deduce that
\[
\Phi ( \overline{v}) =\frac{d^{p( 0) }}{^{p(
0) }}+\frac{d^{p( T) }}{^{p( T) }}
\]
which implies that
\[
\Phi ( \overline{v}) \geq \frac{2}{p^{+}}d^{p_{\ast}}.
\]
Since $d>( \frac{p^{+}}{2}) ^{1/p^{+}}\frac{a}{( T+1)
^{\frac{^{( p^{+}-2) }}{2p^{-}}}( p^{+}) ^{\frac{1}{
p^{-}}}}$, we get $\Phi ( \overline{v}) >r$, where $r$ is as in \eqref{e2.1}.
Moreover, we have
\[
\frac{\Psi ( \overline{v}) }{\Phi ( \overline{v}) }=
\frac{\sum_{k=1}^{T}F(k,d)}{\frac{d^{p( 0) }}{^{p( 0) }
}+\frac{d^{p( T) }}{^{p( T) }}}.
\]
For each $u\in \Phi ^{-1}( ( -\infty ,r) ) $, similarly to \eqref{e2.2}, we have
\[
\frac{1}{r}\sup_{\Phi ( u) \leq r}\Psi ( u) \leq \frac{
l}{r}( T+( T+1) ^{( p^{+}-2) q^{+}/2p^{-}}(
p^{+}) ^{q^{+}/p^{-}}r^{( q/p) _{\ast }}) .
\]
Therefore, from  condition (A4), it holds
\[
\frac{1}{r}\sup_{\Phi ( u) \leq r}\Psi ( u) \leq \frac{
\Psi ( \overline{v}) }{\Phi ( \overline{v}) }.
\]
Hence, Proposition \ref{pro1.4}(1) is verified. Let us proceed with the
coercivity of $I_{\lambda }$. Let $u\in W$ such that
$\| u\|_{W}\to +\infty $. Then, without loss of generality, we can assume
that $\| u\| _{W}>1$. Then from Proposition \ref{pro1.1}(i) and condition (A1),
it reads
\begin{align*}
I_{\lambda }( u)
&=\sum_{k=1}^{T+1}\frac{1}{p(k-1)}|\Delta
u(k-1)|^{p(k-1)}-\lambda \sum_{k=1}^{T}F(k,u( k) ) \\
&\geq \frac{1}{p^{+}(\sqrt{T})^{p^{-}-2}}\| u\|
_{W}^{p^{-}}-T-\lambda l( T+\| u\| _{W}^{q^{+}})
\\
&\geq \frac{1}{p^{+}(\sqrt{T})^{p^{-}-2}}\| u\| _{W}^{p^{-}}-\lambda
l\| u\| _{W}^{q^{+}}-T( 1+\lambda l) ;
\end{align*}
that is, $I_{\lambda }$ is coercive. So, Proposition \ref{pro1.4}(2) is
verified.

Consequently, the assumptions of Proposition \ref{pro1.4} are verified.
Therefore, for each $\lambda \in \Lambda _{r,d}$, the functional 
$I_{\lambda}$ admits at least three distinct critical points that are 
weak solutions of problem \eqref{e1.1}.
\end{proof}

\begin{example} \label{examp3} \rm
As an application of Theorem \ref{the4.1}, if we consider
function $f$ and the assumptions as given in Example \ref{examp1}, 
take $d=4$ and $C_{d}\geq 42$, then the all the assumptions requested 
in Theorem \ref{the4.1} hold. Moreover, for each 
$\lambda \in \Lambda _{r,d}\subseteq ( 25/43,\lambda _a) $  problem 
\eqref{e1.1} admits at least three nontrivial weak solutions, 
where the real constant $\lambda _a$ depends on $a$ and satisfies 
$\lambda_a\geq 31/50$.
\end{example}

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\end{document}