a survey on existence of a solution to fractional difference boundary value problem with |u| p−2u 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 ∈ [1, t]n0 { t +1∇α k (k∇α 0 (u(k))) + k∇α 0 (t +1∇α k (u(k))) + ϕp(u(k)) = λf(k, u(k)), u(0) = u(t + 1) = 0, where 0 < α < 1 and k∇α 0 is the left nabla discrete fractional difference and t +1∇α k is the right nabla discrete fractional difference f : [1, t]n0 × r → r is a continuous function, λ > 0 is a parameter and ϕp is the so called p-laplacian operator defined as ϕp(s) = |s| p−2 s and 1 < p < +∞. the technical method is variational approach for differentiable functionals. several examples are included to illustrate the main results.