SOME PROPERTIES FOR THE RIEMANN–LIOUVILLE FRACTAL CALCULUS OF CONTINUOUS FUNCTIONS

Long Tian ()
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Long Tian: School of Science, Nanjing University of Science and Technology, Nanjing 210094, P. R. China

FRACTALS (fractals), 2021, vol. 29, issue 08, 1-8

Abstract: In this paper, we will prove two properties for the Riemann–Liouville fractal derivatives and integrals, respectively. One is that if f(x) is a Lipschitz function, then f(x) is νth-order differentiable almost everywhere in the Riemann–Liouville sense for any ν ∈ (0, 1). The other is that if the Box dimension for a continuous function f(x) is one, then for any ν ∈ (0, 1), the Box dimension of D−νf(x) is also one. We also give an example to show that there exists a rectifiable function M(x), but for any ν ∈ (0, 1), D−νM(x), the Riemann–Liouville fractal integral of M(x), is rectifiable.

Keywords: Riemann–Liouville Fractal Integral; Riemann–Liouville Fractal Derivative; The Box Dimension; Rectifiable (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21502480

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