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In this paper, we deal with the existence and multiplicity
solutions, for the following fractional  discrete boundary-value problem
begin{equation*}
begin{cases}
_{T+1}nabla_k^{alpha}left( ^{}_knabla_{0}^{alpha}(u(k))right)+{^{}_knabla}_{0}^{alpha}left( ^{}_{T+1}nabla_k^{alpha}(u(k))right)=lambda f(k,u(k)), quad k in [1,T]_{mathbb{N}_{0}},
u(0)= u(T+1)=0,
end{cases}
end{equation*}
where $0leq alphaleq1$ and $^{}_{0}nabla_k^{alpha}$ is  the left nabla discrete fractional difference  and $^{}_knabla_{T+1}^{alpha}$ is the right nabla discrete fractional difference  and   $f: [1,T]_{mathbb{N}_{0}}timesmathbb{R}tomathbb{R}$ is
a continuous function and $lambda>0$ is a parameter. The technical approach is based on the critical point theory and some local
minimum theorems for differentiable functionals. Several examples are included to illustrate the main results.
textbf{MSC(2010):} 26A33; 39A10; 39A27.
textbf{Keywords:}  Discrete fractional calculus, Discrete nonlinear boundary
value problem, Non trivial solution, Variational methods, Critical
point theory. 

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In this paper, we deal with the existence of a non-trivial
solution for the following fractional  discrete boundary-value problem for any $k in [1,T]_{mathbb{N}_{0}}$
begin{equation*}
begin{cases}
_{T+1}nabla_k^{alpha}left( ^{}_knabla_{0}^{alpha}(u(k))right)+{^{}_knabla}_{0}^{alpha}left( ^{}_{T+1}nabla_k^{alpha}(u(k))right)+phi_{p}(u(k))=lambda f(k,u(k)),
 u(0)= u(T+1)=0,
end{cases}
end{equation*}
where $0< alpha<1$ and $^{}_knabla_{0}^{alpha}$ is  the left nabla discrete fractional difference  and $^{}_{T+1}nabla_k^{alpha}$ is the right nabla discrete fractional difference   $f: [1,T]_{mathbb{N}_{0}}timesmathbb{R}tomathbb{R}$ is
a continuous function, $lambda>0$ is a parameter and  $phi _{p}$ is the so called $p$-Laplacian
operator defined as $phi _{p}(s)=|s|^{p-2}s$ and $1