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 νν of the fractional difference satisfies 0<ν≤10<ν≤1. We show that solutions of this IVP satisfy a continuity condition both with respect to the order of the difference, νν, 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.