Continuity of solutions to discrete fractional initial value problems

In this paper, we consider a fractional initial value problem (IVP) in the case where the order of the fractional difference satisfies 0 < 1. We show that solutions of this IVP satisfy a continuity condition both with respect to the order of the difference, , and with respect to the initial conditions, and we deduce several important corollaries from this theorem. Thus, we address a complication that arises in the fractional case but not in the classical (integer-order) case.