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.