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Fractal dimensions of mixed Katugampola fractional integral associated with vector valued functions

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  • Chandra, Subhash
  • Abbas, Syed

Abstract

The aim of this article is to study the fundamental properties of the mixed Katugampola fractional integral (K-integral) of vector-valued functions and fractal dimensional results. We show that the mixed K-integral preserves the basic properties such as boundedness, continuity, and bounded variation of vector-valued functions. We also estimate the Hausdorff dimension of the graph of the vector-valued function and the graph of the mixed K-integral on the rectangular region. Moreover, we prove that the upper bound of the box dimension of the graph of each coordinate function of mixed K-integral of vector-valued functions is 3−min{μ1,μ2}, where μ1 and μ2 are order of the fractional integral with 0<μ1<1,0<μ2<1. Moreover, we give an example of unbounded variational vector-valued functions. In the end, we discuss some problems for future direction.

Suggested Citation

  • Chandra, Subhash & Abbas, Syed, 2022. "Fractal dimensions of mixed Katugampola fractional integral associated with vector valued functions," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    • Handle: RePEc:eee:chsofr:v:164:y:2022:i:c:s096007792200827x
    DOI: 10.1016/j.chaos.2022.112648
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    References listed on IDEAS

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    1. Yao, K. & Liang, Y.S. & Zhang, F., 2009. "On the connection between the order of the fractional derivative and the Hausdorff dimension of a fractal function," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2538-2545.
    2. Wei Xiao, 2022. "On Box Dimension Of Hadamard Fractional Integral (Partly Answer Fractal Calculus Conjecture)," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(04), pages 1-10, June.
    3. Liang, Yongshun, 2009. "On the fractional calculus of Besicovitch function," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2741-2747.
    4. Subhash Chandra & Syed Abbas, 2021. "The Calculus Of Bivariate Fractal Interpolation Surfaces," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(03), pages 1-13, May.
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    Cited by:

    1. Yu, Binyan & Liang, Yongshun, 2024. "On two special classes of fractal surfaces with certain Hausdorff and Box dimensions," Applied Mathematics and Computation, Elsevier, vol. 468(C).
    2. Yu, Binyan & Liang, Yongshun, 2024. "On the dimensional connection between a class of real number sequences and local fractal functions with a single unbounded variation point," Chaos, Solitons & Fractals, Elsevier, vol. 183(C).
    3. Verma, Manuj & Priyadarshi, Amit, 2023. "Graphs of continuous functions and fractal dimensions," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    4. Lal, Rattan & Chandra, Subhash & Prajapati, Ajay, 2024. "Fractal surfaces in Lebesgue spaces with respect to fractal measures and associated fractal operators," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).

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