Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions

In this paper, we consider a discrete fractional boundary value problem of the form −Δνy(t) = f (t + ν − 1, y(t + ν − 1)), y(ν − 2) = g(y), y(ν + b) = 0, where f : [ν − 1, . . . , ν+b−1]Nν−2 ×R → R is continuous, g : C([ν−2, ν+b]Nν−2 , R) is a given functional, and 1 < ν ≤ 2. We give a representation for the solution to this problem. Finally, we prove the existence and uniqueness of solution to this problem by using a variety of tools from nonlinear functional analysis including the contraction mapping theorem, the Brouwer theorem, and the Krasnosel’skii theorem.