Mathematics, Department of
Date of this Version
6-17-2015
Abstract
In this article, we are concerned with the relationships between the sign of Caputo fractional differences and integer nabla differences. In particular, we show that if N -1 < v < N. f : Na -N + 1 -> R, va * f(t) > O, for t - Na +1 and N-1f(a) > 0, then N -1 f(t) > 0 for t- Na +1, then va* f(t) > 0, for each t - Na +1. As applications of these two results, we get that if 1 < vR, va*f(t) > 0 for t - Na +1 and f(a) > f(a-1), then f(t) is an increasing function for t- Na -1. Conversely if 0 < vR and f is an increasing function for t - Na, then va*f(t) > 0, for each t - Na +1. We also give a counterexample to show that the above assumption f(a) > f(a-1) in the last result is essential. These results demonstrate that, in some sense, the positivity of the v-th order Caputo fractional difference has a strong connection to the monotonicity of f(t).