Advances in Analysis

Download PDF (461.4 KB) PP. 121 - 128 Pub. Date: March 9, 2017

DOI: 10.22606/aan.2017.22006

• Yang Li
  Institute of Science, Nanjing University of Science and Technology, Nanjing, 210094, China
• Yongshun Liang^*
  
  Institute of Science, Nanjing University of Science and Technology, Nanjing, 210094, China

In the present paper, upper bound estimation of upper Box dimension of Riemann- Liouville fractional integral of order ν of any continuous functions on a closed interval has been proved to be no more than 2 − ν when 0 < ν < 1. If a continuous function which satisfies -Hölder condition on a closed interval, upper Box dimension of its Riemann-Liouville fractional integral is no more than 2 − α when 0 < α < 1. Upper bound of upper Box dimension of Riemann-Liouivlle fractional integral of certain type of fractal functions has been proved to be no more than Box dimension of functions themselves.

Box dimension, Riemann-Liouville fractional integral, fractal function, upper bound, length of graph.

[1] J. Falconer, Fractal geometry: mathematical foundations and applications, John Wiley Sons Inc., Newyork. 1990.

[2] Y. S. Liang, The relationship between the Box dimension of the Besicovitch functions and the orders of their fractional calculus, Applied Mathematics and Computation 200 (2008), 197-207.

[3] Y. S. Liang, On the fractional calculus of Besicovitch function, Chaos, Solitons and Fractals 42 (2009), 2741-2747.

[4] Y. S. Liang, Box dimensions of Riemann-Liouville fractional integrals of continuous functions of bounded variation, Nonlinear Analysis 72 (2010), 4304-4306.

[5] Y. S. Liang, Some remarks on continuous functions of unbounded variation. Acta Mathematica Sinica, Chinese Series 59 (2016), 215-232.

[6] K. B. Oldham, J. Spanier, The fractional calculus, Academic Press, Newyork, 1974.

[7] B. Ross, The fractional calculus and its applications, Springer-Verlag, Berlin, Heidelberg, 1975.

[8] H. J. Ruan, W. Y. Su, K. Yao, Box dimension and fractional integral of linear fractal interpolation functions, Journal of Approximation Theory 161 (2009), 187-197.

[9] F. B. Tatom, The relationship between fractional calculus and fractals, Fractals 3 (1995), 217-229.

[10] Z. Y. Wen, Mathematical foundations of fractal geometry, Science Technology Education Publication House, Shanghai, 2000[in Chinese].

[11] K. Yao, W. Y. Su, S. P. Zhou, On the fractional calculus of a type of Weierstrass function, Chinese Annals of Mathematics 25(A) (2004), 711-716.

[12] K. Yao, W. Y. Su, S. P. Zhou, On the fractional derivatives of a fractal function, Acta Mathematica Sinica, English Series 22 (2006), 719-722.

[13] M. Z?hle, H. Ziezold, Fractional derivatives of Weierstrass-type functions, Journal of Computational and Applied Mathematics 76 (1996), 265-275.

[14] S. P. Zhou, Fractional integrals of the Weierstrass functions: The exact Box dimension, Analysis in Theory and Applications. 20 (2004), 332-241.