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.