Volume 4, Issue 1 (6-2023)
MACO 2023, 4(1): 45-59 |
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Abstract: (222 Views)
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
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
Keywords: Discrete fractional calculus, Discrete nonlinear boundary value problem, Non trivial solution, Variational methods, Critical point theory.
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