On Box Dimension Of Hadamard Fractional Integral (Partly Answer Fractal Calculus Conjecture)

This paper probes into change of box dimension for an arbitrary fractal continuous function after Hadamard fractional integration. For classic calculus, we know that a function’s differentiability increases one-order after integration, and decreases one-order after differentiation. But for fractional integro-differentiation, this similar fundamental problem is still open [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals 2 (1995) 217–229; M. Zähle and H. Ziezold, Fractional derivatives of Weierstrass-type functions, J. Comput. Appl. Math. 76 (1996) 265–275; Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of 1-dimensional continuous functions, Sci. Sin. Math. 4 (2016) 423–438 (in Chinese); Y. S. Liang, Progress on estimation of fractal dimensions of fractional calculus of continuous functions, Fractals 26 (2019), doi:10.1142/S0218348X19500841] (see conjecture 1.1), although it is believed that the roughness of a fractal function is increased or decreased accounts α after α-order fractional differentiation or integration. This paper partly answers this problem. Particularly, the estimation of box dimension of Hadamard fractional integral is of methodology for considering similar fractional integration. In this paper, it is proved that the upper box dimension of the graph of f(x) does not increase after α-order Hadamard fractional integral DH−αf(x), i.e. dim¯BΥ(DH−αf,I) ≤dim¯ BΥ(f,I). Particularly, while dimBΥ(f,I) = 1, for example, a function is of unbounded variation and/or infinite length, it holds dimBΥ(DH−αf,I) =dim BΥ(f,I) = 1, which answers completely this conjecture for one-dimensional fractal function. The published paper could verify Conjecture 1.1 under Riemann–Liouville or Wyel fractional integral, for only constructed functions with one-dimensional fractal [Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of 1-dimensional continuous functions, Sci. Sin. Math. 4 (2016) 423–438 (in Chinese); Y. Li and W. Xiao, Fractal dimensions of Riemann–Liouville fractional integral of certain unbounded variational continuous function, Fractals 25 (2017), doi:10.1142/S0218348X17500475; Y. S. Liang, Fractal dimension of Riemann–Liouville fractional integral of 1-dimensional continuous functions, Fract. Calc. Appl. Anal. 21 (2018) 1651–1658; Q. Zhang, Some remarks on one-dimensional functions and their Riemann–Liouville fractional calculus, Acta Math. Sin. 3 (2014) 517–524; J. Wang and K. Yao, Construction and analysis of a special one-dimensional continuous function, Fractals 25 (2017), doi:10.1142/S0218348X17500207; X. Liu, J. Wang and H. L. Li, The classification of one-dimensional continuous functions and their fractional integral, Fractals 26 (2018), doi:10.1142/S0218348X18500639].