A note on Katugampola fractional calculus and fractal dimensions

Abstract

The goal of this paper is to study the Katugampola fractional integral of a continuous function of bounded variation defined on a closed bounded interval. We note that the Katugampola fractional integral of a function shares some analytical properties such as boundedness, continuity and bounded variation of the function defining it. Consequently, we deduce that fractal dimensions – Minkowski dimension and Hausdorff dimension – of the graph of the Katugampola fractional integral of a continuous function of bounded variation are one. A natural question then arises is whether there exists a continuous function which is not of bounded variation with its graph having fractal dimensions one. In the last part of the article, we construct a continuous function, which is not of bounded variation and for which the graph has fractal dimensions one. The construction enunciated herein includes previous constructions found in the recent literature as special cases. The article also hints at an upper bound for the upper box dimension of the graph of the Katugampola fractional derivative of a continuously differentiable function.