Revista de la
Unión Matemática Argentina
A note on the Schwarz fractal derivative
Luis Ángel García Pacheco, Daniel Alfonso Santiesteban, Ricardo Abreu Blaya, and José María Sigarreta Almira

Volume 69, no. 1 (2026), pp. 1–20    

Published online (final version): November 19, 2025

https://doi.org/10.33044/revuma.4981

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Abstract

We define a Schwarz fractal derivative of order $n$ for a real-valued function $f(t)$ as the limit \[ (\mathrm{S}^{\alpha,\beta}_nf)(t_0)=\lim_{t\to t_0} \frac{\sum_{j=0}^n\binom{n}{j}(-1)^jf^\beta\bigl(t_0+\frac{n-2j}{2}(t-t_0)\bigr)}{(t^\alpha-t_0^\alpha)^n}, \] where $\alpha,\beta>0$ and $f^\beta:=f|f|^{\beta-1}$. This derivative naturally generalizes the one introduced by Riemann in 1854. We study its essential properties and its relationship with other fractal derivatives recently reported in the literature. We obtain certain analogues of the mean value and Rolle theorems, together with some of their most important consequences. Finally, we propose an extension of such derivatives to the several-variable setting.

References

  1. D. Alfonso Santiesteban, A. Portilla, J. M. Rodríguez-García, and J. M. Sigarreta, On fractal derivatives and applications, Math. Methods Appl. Sci. 48 no. 11 (2025), 10726–10739.  DOI  MR  Zbl
  2. J. M. Ash, A new, harder proof that continuous functions with Schwarz derivative $0$ are lines, in Fourier analysis: Analytic and geometric aspects (Orono, ME, 1992), Lecture Notes in Pure and Appl. Math. 157, Dekker, New York, 1994, pp. 35–46.  MR  Zbl
  3. A. Atangana and S. Qureshi, Modeling attractors of chaotic dynamical systems with fractal-fractional operators, Chaos Solitons Fractals 123 (2019), 320–337.  DOI  MR  Zbl
  4. C. E. Aull, The first symmetric derivative, Amer. Math. Monthly 74 (1967), 708–711.  DOI  MR  Zbl
  5. C. L. Belna, M. J. Evans, and P. D. Humke, Symmetric and ordinary differentiation, Proc. Amer. Math. Soc. 72 no. 2 (1978), 261–267.  DOI  MR  Zbl
  6. W. Chen, Time-space fabric underlying anomalous diffusion, Chaos Solitons Fractals 28 no. 4 (2006), 923–929.  DOI  Zbl
  7. F. M. Filipczak, Sur les dérivées symétriques des fonctions approximativement continues, Colloq. Math. 34 no. 2 (1976), 249–256.  DOI  MR  Zbl
  8. A. K. Golmankhaneh, Fractal calculus and its applications: $\mathrm{F}^\alpha$-calculus, World Scientific, Singapore, 2022.  DOI  MR  Zbl
  9. S. L. Haines, The symmetric derivative, Master's thesis, Bowling Green State University, 1965. Available at http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1670603448495953.
  10. J. C. Hernández-Gómez, R. Reyes, J. M. Rodríguez, and J. M. Sigarreta, Fractional model for the study of the tuberculosis in Mexico, Math. Methods Appl. Sci. 45 no. 17 (2022), 10675–10688.  DOI  MR  Zbl
  11. R. Kanno, Representation of random walk in fractal space-time, Phys. A 248 no. 1-2 (1998), 165–175.  DOI
  12. L. Larson, The symmetric derivative, Trans. Amer. Math. Soc. 277 no. 2 (1983), 589–599.  DOI  MR  Zbl
  13. S. Rădulescu, P. Alexandrescu, and D.-O. Alexandrescu, Generalized Riemann derivative, Electron. J. Differential Equations 2013 (2013), No. 74.  MR  Zbl Available at https://emis.de/ft/1075.
  14. B. Riemann, Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe, Dieterich, Göttingen, 1867. Available at https://eudml.org/doc/203787.
  15. M. Spivak, Calculus, second ed., Publish or Perish, Berkeley, CA, 1980.  Zbl
  16. E. M. Stein and A. Zygmund, On the differentiability of functions, Studia Math. 23 (1964), 247–283.  DOI  MR  Zbl