A survey on existence of a solution to fractional difference boundary value problem with $|u|^{p-2}u$ term

In this paper, we deal with the existence of a non-trivialsolution 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   $f: [1,T]_{mathbb{N}_{0}}timesmathbb{R}tomathbb{R}$ isa continuous function, $lambda>0$ is a parameter and  $phi _{p}$ is the so called $p$-Laplacianoperator defined as $phi _{p}(s)=|s|^{p-2}s$ and $1