A survey on existence of a solution to fractional difference boundary value problem with |u|^{p-۲}u term

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 [۱,T]_{mathbb{N}_{۰}} begin{equation*} begin{cases} _{T+۱}nabla_k^{alpha}left( ^{}_knabla_{۰}^{alpha}(u(k))right)+{^{}_knabla}_{۰}^{alpha}left( ^{}_{T+۱}nabla_k^{alpha}(u(k))right)+phi_{p}(u(k))=lambda f(k,u(k)),  u(۰)= u(T+۱)=۰, end{cases} end{equation*} where ۰< alpha<۱ and ^{}_knabla_{۰}^{alpha} is  the left nabla discrete fractional difference  and ^{}_{T+۱}nabla_k^{alpha} is the right nabla discrete fractional difference   f: [۱,T]_{mathbb{N}_{۰}}timesmathbb{R}tomathbb{R} is a continuous function, lambda>۰ is a parameter and  phi _{p} is the so called p-Laplacian operator defined as phi _{p}(s)=|s|^{p-۲}s and ۱

کلیدواژه ها:

Discrete fractional calculus ، Discrete nonlinear boundary value problem ، Non trivial solution ، Variational methods ، Critical point theory.