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BOX DIMENSION OF HADAMARD FRACTIONAL INTEGRAL OF CONTINUOUS FUNCTIONS OF BOUNDED AND UNBOUNDED VARIATION

XIAO ER WU

Institute of Science, Nanjing University of Science and Technology, Nanjing 210094, P. R. China

E-mail Address: [email protected]

Corresponding author.

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JUN HUAI DU

Dazhou Hydrowater School in Sichuan Province, Dazhou 635000, P. R. China

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https://doi.org/10.1142/S0218348X17500359Cited by:19 (Source: Crossref)

Abstract

The present paper investigates fractal dimension of Hadamard fractional integral of continuous functions of bounded and unbounded variation. It has been proved that Hadamard fractional integral of continuous functions of bounded variation still is continuous functions of bounded variation. Definition of an unbounded variation point has been given. We have proved that Box dimension and Hausdorff dimension of Hadamard fractional integral of continuous functions of bounded variation are $1$ . In the end, Box dimension and Hausdorff dimension of Hadamard fractional integral of certain continuous functions of unbounded variation have also been proved to be $1$ .

Keywords:

References

1. Y. S. Liang and W. Y. Su, The relationship between the fractal dimensions of a type of fractal functions and the order of their fractional calculus, Chaos Solitons Fractals 34 (2007) 682–692. Crossref, Web of Science, Google Scholar
2. K. M. Kolwankar and A. D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos 6 (1996) 505–513. Crossref, Web of Science, Google Scholar
3. M. Zähle and H. Ziezold, Fractional derivatives of Weierstrass-type functions, J. Comput. Appl. Math. 76 (1996) 265–275. Crossref, Web of Science, Google Scholar
4. F. B. Tatom, The relationship between fractional calculus and fractals, Fractals 3 (1995) 217–229. Link, Web of Science, Google Scholar
5. K. Yao and W. Y. Su, On the connection between the order of the fractional calculus and the dimension of a fractal function, Chaos Solitons Fractals 23 (2005) 621–629. Crossref, Web of Science, Google Scholar
6. M. F. Dai, Lipschitz equivalence of general sierpinski carpets, Fractals 16 (2008) 379–388. Link, Web of Science, Google Scholar
7. K. Yao, W. Y. Su and S. P. Zhou, On the fractional calculus of a type of Weierstrass function, Chin. Ann. Math. 25 (2004) 711–716. Google Scholar
8. Y. S. Liang, Box dimensions of Riemann–Liouville fractional integrals of continuous functions of bounded variation, Nonlinear Anal. 72 (2010) 4304–4306. Crossref, Web of Science, Google Scholar
9. Y. S. Liang, Fractal dimension of certain continuous functions of unbounded variation, Fractals 25 (2017) 1750009, doi: 10.1142/S0218348X17500098. Link, Web of Science, Google Scholar
10. J. Wang and K. Yao, Dimension analysis of continuous functions with unbounded variation, Fractals 25 (2017) 1730001, doi: 10.1142/S0218348X1730001X. Link, Web of Science, Google Scholar
11. H. J. Ruan, W. Y. Su and K. Yao, Box dimension and fractional integral of linear fractal interpolation functions, J. Approx. Theory 161 (2009) 187–197. Crossref, Web of Science, Google Scholar
12. Y. S. Liang, A type of fractal interpolation functions and their fractional calculus, Fractals 24 (2016) 1650026, doi: 10.1142/S0218348X16500262. Link, Web of Science, Google Scholar
13. F. Falconer, Fractal Geometry: Mathematical Foundations and Applications (John Wiley Sons Inc., New York, 1990). Google Scholar
14. K. S. Millker and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (John Wiley Sons Inc., New York, 1993). Google Scholar
15. W. X. Zheng and S. W. Wang, Real Function and Functional Analysis (High Education Publication, Beijing, 1980) (in Chinese). Google Scholar
16. Q. Zhang, Some remarks on one-dimensional functions and their Riemann–Liouville fractional calculus, Acta Math. Sin. Engl. Ser. 30 (2014) 517–524. Crossref, Web of Science, Google Scholar
17. Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of 1-dimensional continuous functions, Sci. China Math. Chin. Ser. 46 (2016) 423–438. Google Scholar