Nonsymmetric and symmetric fractional calculi on arbitrary nonempty closed sets

Abstract

We introduce a nabla, a delta, and a symmetric fractional calculus on arbitrary nonempty closed subsets of the real numbers. These fractional calculi provide a study of differentiation and integration of non-integer order on discrete, continuous, and hybrid settings. Main properties of the new fractional operators are investigated and some fundamental results presented, illustrating the interplay between discrete and continuous behaviors. © 2015 John Wiley & Sons, Ltd.