A NOTE ON FRACTAL DIMENSION OF RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL

Abstract

This paper intends to study the analytical properties of the Riemann–Liouville fractional integral and fractal dimensions of its graph on [Formula: see text]. We show that the Riemann–Liouville fractional integral preserves some analytical properties such as boundedness, continuity and bounded variation in the Arzelá sense. We also deduce the upper bound of the box dimension and the Hausdorff dimension of the graph of the Riemann–Liouville fractional integral of Hölder continuous functions. Furthermore, we prove that the box dimension and the Hausdorff dimension of the graph of the Riemann–Liouville fractional integral of a function, which is continuous and of bounded variation in Arzelá sense, are [Formula: see text].