A NEW ESTIMATION OF BOX DIMENSION OF RIEMANN–LIOUVILLE FRACTIONAL CALCULUS OF CONTINUOUS FUNCTIONS

Abstract

This paper establishes a linear relationship between the order of the Riemann–Liouville fractional calculus and the exponent of the Hölder condition, whether the Hölder condition is global, local, or at a single point. We propose and prove a control inequality between the Hölder derivative ([Formula: see text] as defined in Proposition 12) of a continuous function and the Hölder derivative of the Riemann–Liouville fractional calculus of this function. In addition, this paper provides a more accurate estimation of the Box dimension of the graph of the Riemann–Liouville fractional integral of an arbitrary continuous function. More specifically, it establishes the result that whenever there is a continuous function whose graph has the upper Box dimension [Formula: see text] with [Formula: see text], the graph of its Riemann–Liouville fractional integral of order [Formula: see text], with [Formula: see text], has the upper Box dimension not greater than [Formula: see text].