THE ESTIMATES OF HÖLDER INDEX AND THE BOX DIMENSION FOR THE HADAMARD FRACTIONAL INTEGRAL

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

FRACTALS (fractals), 2021, vol. 29, issue 03, 1-12

Abstract: This paper focuses on the Hölder continuity and the Box Dimension to the pth Hadamard Fractional Integral (HFI) on a given interval [α,β]. We use Hpϕ(x) to denote it. In this paper, two different methods are used to study this problem. By using the approximation method, we obtain that for ϕ(x) ∈ℋq([α,β]) with p ∈ (0, 1) and q ∈ (0, 1 − p), if α > 0, then Hpϕ(x) is γth Hölder continuous in (α,β] with γ = p (1−q)(1+p)−p2, and is p 1+pth Hölder continuous on [α,β]. Moreover, the Box Dimension of the graph of Hpϕ(x) on the interval [α,β] is less than or equal to 2 − (1+2p)γ (1+p)γ+(1+2p). If α = 0, then Hpϕ(x) is locally γth Hölder continuous in (0,β] with the same γ, and the Box Dimension of Hpϕ(x) on [0,β] is less than or equal to 2 − γ 1+γ. By using another method, we imply that, for ϕ(x) ∈ℋq([α,β]) with q ∈ (0, 1 − p) and 0 < p < 1, if α > 0, then Hpϕ(x) is (p + q)th Hölder continuous, and thus the Box Dimension of the graph of Hpϕ(x) is no more than 2 − (p + q); if α = 0, then Hpϕ(x) is locally (p + q)th Hölder continuous in (0,β], and is qth Hölder continuous at 0. Then the Box Dimension of the graph to Hpϕ(x) on [0,β] is less than or equal to 2 −(1+q)(p+q) 1+2q+p. We also give two examples to show that the above Hölder indexes given by the second method are optimal.

Keywords: The Hadamard Fractional Integral; The Hölder Index; The Box Dimension (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21500729

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