On a fractional boundary value problem with fractional boundary conditions

In this paper, we consider a discrete fractional boundary value problem, for t ∈ [ 0 , b + 1 ] N 0 , of the form − Δ ν y ( t ) = f ( t + ν − 1 , y ( t + ν − 1 ) ) , y ( ν − 2 ) = 0 , [ Δ α y ( t ) ] t = ν + b − α + 1 = 0 , where f : [ ν − 1 , … , ν + b ] N ν − 2 × R → R is continuous, 1 < ν ≤ 2 , and 0 ≤ α < 1 . We prove that this problem can be interpreted as a discrete multipoint problem. We also show that the problem is a generalization of some recent results. Our results provide some basic analysis of discrete fractional boundary conditions.