ON THE BOX DIMENSION OF WEYL–MARCHAUD FRACTIONAL DERIVATIVE AND LINEARITY EFFECT

Abstract

This paper intends to estimate the box dimension of the Weyl–Marchaud fractional derivative (Weyl–M derivative) for various choices of continuous functions on a compact subset of [Formula: see text] We show that the Weyl–M derivative of order [Formula: see text] of a continuous function satisfying Hölder condition of order [Formula: see text] also satisfies Hölder condition of order [Formula: see text] and the upper box dimension of the Weyl–M derivative increases at most linearly with the order [Formula: see text]. Moreover, the upper box dimension of the Weyl–M derivative of a continuous function satisfying the Lipschitz condition is not more than the sum of the box dimension of the function itself and order [Formula: see text]. Furthermore, we prove that the box dimension of the Weyl–M derivative of a certain continuous function which is of bounded variation is one.