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Fractal dimensions of fractional integral of continuous functions

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Abstract

In this paper, we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals. Riemann–Liouville integral of a continuous function f(x) of order v(v > 0) which is written as D −v f(x) has been proved to still be continuous and bounded. Furthermore, upper box dimension of D −vf(x) is no more than 2 and lower box dimension of D −v f(x) is no less than 1. If f(x) is a Lipshciz function, D −v f(x) also is a Lipshciz function. While f(x) is differentiable on [0, 1], D −v f(x) is differentiable on [0, 1] too. With definition of upper box dimension and further calculation, we get upper bound of upper box dimension of Riemann–Liouville fractional integral of any continuous functions including fractal functions. If a continuous function f(x) satisfying Hölder condition, upper box dimension of Riemann–Liouville fractional integral of f(x) seems no more than upper box dimension of f(x). Appeal to auxiliary functions, we have proved an important conclusion that upper box dimension of Riemann–Liouville integral of a continuous function satisfying Hölder condition of order v(v > 0) is strictly less than 2 − v. Riemann–Liouville fractional derivative of certain continuous functions have been discussed elementary. Fractional dimensions of Weyl–Marchaud fractional derivative of certain continuous functions have been estimated.

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Authors and Affiliations

  1. Institute of Science, Nanjing University of Science and Technology, Nanjing, 210014, P. R. China

    Yong Shun Liang

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China

    Wei Yi Su

Authors
  1. Yong Shun Liang
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  2. Wei Yi Su
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Correspondence to Yong Shun Liang.

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Supported by National Natural Science Foundation of China (Grant Nos. 11201230 and 11271182)

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Liang, Y.S., Su, W.Y. Fractal dimensions of fractional integral of continuous functions. Acta. Math. Sin.-English Ser. 32, 1494–1508 (2016). https://doi.org/10.1007/s10114-016-6069-z

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  • DOI: https://doi.org/10.1007/s10114-016-6069-z

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