Abstract
We extend to s-dimensional fractal sets the notion of first return integral (Definition 5) and we prove that there are s-derivatives not s-first return integrable.
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Bongiorno, B.: On the first-return integrals. J. Math. Anal. Appl. 333, 112–116 (2007)
Bongiorno, D.: Riemann-type definition of the improper integrals. Czech. Math. J. 54, 717–725 (2004)
Bongiorno, D., Corrao, G.: On the Fundamental theorem of Calculus for fractal sets. Fractals 23(2), 1550008 (2015)
Darji, U.B., Evans, M.J.: A first-return examination of the Lebesgue integral. Real Anal. Exch. 27, 578–581 (2001–2002)
Darji, U.B., Evans, M.J.: Functions not first-return integrable. J. Math. Anal. Appl. 347, 381–390 (2008)
Jiang, H., Su, W.: Some fundamental results of calculus on fractal sets. Commun. Nonlinear Sci. Numer. Simul. 3(1), 22–26 (1998)
Parvate, A., Gangal, A.D.: Calculus on fractal subsets of real line I. Formulation. Fractals 17(1), 53–81 (2009)
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)
De Guzman, M., Martin, M.A., Reyes, M.: On the derivation of Fractal functions. In: Proceedings of the 1st IFIT Conference on Fractals in the Fundamental and Applied Sciences, Lisbon, 6–8 June 1990, North-Holland, pp. 169–182 (1991)
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Communicated by P. De Lucia.
Dedicated to Prof. Hans Weber on the occasion of his 70th birthday.
Supported by GNAMPA- INDAM of Italy.
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Bongiorno, D. Derivatives not first return integrable on a fractal set. Ricerche mat 67, 597–604 (2018). https://doi.org/10.1007/s11587-018-0390-z
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DOI: https://doi.org/10.1007/s11587-018-0390-z