A survey on multiplicity results for fractional difference equations and variational method

In this paper, we deal with the existence and multiplicity solutions, for the following fractional discrete boundary-value problem { T +1∇α k (k∇α 0 (u(k))) + k∇α 0 (T +1∇α k (u(k))) = λf(k, u(k)), k ∈ [1, T]N0 , u(0) = u(T + 1) = 0, where 0 ≤ α ≤ 1 and 0∇α k is the left nabla discrete fractional difference and k∇α T +1 is the right nabla discrete fractional difference and f : [1, T]N0 × R → R is a continuous function and λ > 0 is a parameter. The technical approach is based on the critical point theory and some local minimum theorems for differentiable functionals. Several examples are included to illustrate the main results.