Derivatives not first return integrable on a fractal set

  • Donatella Bongiorno


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.


s-dimensional Hausdorff measure s-set s-derivative Henstock–Kurzweil integral First return integral 

Mathematics Subject Classification

26A39 26A42 28A80 


  1. 1.
    Bongiorno, B.: On the first-return integrals. J. Math. Anal. Appl. 333, 112–116 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bongiorno, D.: Riemann-type definition of the improper integrals. Czech. Math. J. 54, 717–725 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bongiorno, D., Corrao, G.: On the Fundamental theorem of Calculus for fractal sets. Fractals 23(2), 1550008 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Darji, U.B., Evans, M.J.: A first-return examination of the Lebesgue integral. Real Anal. Exch. 27, 578–581 (2001–2002)Google Scholar
  5. 5.
    Darji, U.B., Evans, M.J.: Functions not first-return integrable. J. Math. Anal. Appl. 347, 381–390 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jiang, H., Su, W.: Some fundamental results of calculus on fractal sets. Commun. Nonlinear Sci. Numer. Simul. 3(1), 22–26 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Parvate, A., Gangal, A.D.: Calculus on fractal subsets of real line I. Formulation. Fractals 17(1), 53–81 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  9. 9.
    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)Google Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.Dipartimento Energia, Ingegneria dell’Informazione e Modelli MatematiciUniversità di PalermoPalermoItaly

Personalised recommendations