Analytical properties of (k, s)-Riemann–Liouville fractional integral and its fractal dimension

Abstract

A prominent study problem that gained unbearable attraction is to calculate the fractal dimensions of the graph of a fractional integral of a function. The present article is a study about the analytical properties, such as boundedness, continuity and bounded variation of (k, s)-Riemann–Liouville fractional integral of a function, which generalizes the Katugampola fractional integral. Further, we investigate the Hausdorff dimension and box dimension of the graph of (k, s)-Riemann–Liouville fractional integral of a continuous function defined on a compact interval [a, b] and derived the conclusion that these two dimensions are 1. Also we establish a relation between the order of the fractional integral and upper box dimension.