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.