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Based on recent variational methods for smooth functionals defined on reflexive Banach spaces, We prove the existence of at least one
non-trivial solution for a class of  p-Hamiltonian systems. Employing one critical point theorem, existence of at least one weak solutions is ensured. This approach is based on variational methods and critical point theory. The technical approach is mainly based on the at least one non -trivial solution critical point theorem of G. Bonanno.

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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 study the existence of at least three distinct
solutions for a class of impulsive fractional boundary value
problems with $p$-Laplacian with Dirichlet boundary conditions.
Our approach is based on recent variational methods for smooth
functionals defined on reflexive Banach spaces. One example is
presented to demonstrate the application of our main results.

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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

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The existence of infinitely many solutions for a class of impulsive
fractional boundary value problems is established.
Our approach is based on recent variational methods for smooth
functionals defined on reflexive Banach spaces. Some recent results are extended and improved. One example is
given in this paper to illustrate the main results.