Weakly absolutely continuous functions without weak, but fractional weak derivatives

  • Hussein A. H. SalemEmail author


Let E be an infinite-dimensional Banach space and I be a compact interval of the real line. The aim of this paper is two-fold: On the one hand, we construct an example of a weakly absolutely continuous function taking its values in E that is nowhere weakly differentiable on I, but has weakly continuous fractional weak derivatives of some critical orders less than one. This also holds for (nearly) all orders less than one if E failing cotype. We believe that this results are of independent interest and discuss it in a rather general setting. On the other hand, we establish some examples of weakly continuous functions taking its values in Gauge space fail to be pseudo differentiable on I, but have fractional-pseudo derivatives of “all” order less than one. An application will be given.


Fractional calculus Orlicz spaces Pettis integrals 

Mathematics Subject Classification

26A33 34G20 



  1. 1.
    Agarwal, R.P., Lupulescu, V., O’Regan, D., Rahman, G.: Weak solutions for fractional differential equations in nonreflexive Banach spaces via Riemann-Pettis integrals. Math. Nachr. 289(4), 395–409 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agarwal, P., Al-Mdalla, Q., Cho, Y.J.E., Jain, S.: Fractional differential equations for the generalized Mittag-Leffler function. Adv. Differ. Equ. 2018, 58 (2018). MathSciNetCrossRefGoogle Scholar
  3. 3.
    Agarwal, P., El-Sayed, A.A.: Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation. Physica A Stat. Mech. Appl. 500, 40–49 (2018). MathSciNetCrossRefGoogle Scholar
  4. 4.
    Baltaeva, U., Agarwa, P.: Boundary-value problems for the third-order loaded equation with noncharacteristic type-change boundaries. Math. Methods Appl. Sci. 41(9), 3307–3315 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bartle, R.G.: A Modern Theory of Integration, Graduate Studies in Mathematics. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  6. 6.
    Benchohra, M., Mostefai, F.: Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach spaces. Opuscula Math. 32(1), 31–40 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Calabuig, J.M., Rodríguez, J., Rueda, P., Sánchez-Pérez, E.A.: On \( p \)-Dunford integrable functions with values in Banach spaces. J. Math. Anal. Appl. 464(1), 806–822 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Diestel, J., Uhl Jr., J.J.: Vector Measures, Mathematical Surveys, vol. 15. American Mathematical Society, Providence (1977)CrossRefzbMATHGoogle Scholar
  9. 9.
    Dilworth, J., Girardi, M.: Nowhere weak differentiability of the Pettis integral. Quaest. Math. 18(4), 365–380 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Munroe, M.E.: A note on weak differentiability of Pettis integrals. Bull. Am. Math. Soc. 52, 167–174 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jainm, S., Agarwal, P., Kilicman, A.: Pathway fractional integral operator associated with 3m-parametric Mittag–Leffter functions. Int. J. Appl. Comput. Math. 4, 115 (2018)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kadets, V.: Non-differentiable indefinite Pettis integrals. Quaest. Math. 17, 137–149 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mendel, M., Naor, A.: Metric cotype. Ann. Math. 168, 247–298 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Naralenkov, K.: On Denjoy type extension of the Pettis integral. Czechoslov. Math. J. 60(135), 737–750 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pettis, B.J.: On integration in vector spaces. Trans. Am. Math. Soc. 44, 277–304 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Phillips, R.S.: Integration in a convex linear topological space. Trans. Am. Math. Soc. 47, 114–145 (1940)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ross, B., Samko, S.G., Love, E.R.: Functions that have no first order derivative might have fractional derivative of all orders less than one. Real Anal. Exch. 20, 140–157 (1994/1995)Google Scholar
  18. 18.
    Salem, H.A.H.: On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies. J. Comput. Appl. Math. 224, 565–572 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Salem, H.A.H.: On the fractional calculus in abstract spaces and their applications to the Dirichlet-type problem of fractional order. Comput. Math. Appl. 59, 1278–1293 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Salem, H.A.H., Cichoń, M.: On solutions of fractional order boundary value problems with integral boundary conditions in Banach spaces. J. Funct. Spaces Appl. 13 (Article ID 428094) (2013)Google Scholar
  21. 21.
    Salem, H.A.H.: Hadamard-type fractional calculus in Banach spaces, Revista de la Real Academia de Ciencias Exactas, Fsicas y Naturales. Serie A. Matemticas (2018)
  22. 22.
    Salem, H.A.H.: On functions without pseudo derivatives having fractional pseudo derivatives. Quaest. Math. (2018),
  23. 23.
    Salem, H.A.H.: On the theory of fractional calculus in the Pettis-function spaces. J. Funct. Spaces Appl. 13 (Article ID 8746148) (2018)Google Scholar
  24. 24.
    Solomon, D.: Denjoy Integration in Abstract Spaces, Memories of the American Mathematical Society. American Mathematical Society, Providence (1969)Google Scholar
  25. 25.
    Szep, A.: Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces. Studia Sci. Math. Hungar. 6, 197–203 (1971)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Yeong, L.T.: Henstock–Kurzweil Integration on Euclidean Spaces, vol. 12. World Scientific Publishing Co. Pte. Ltd, Singapore (2011)zbMATHGoogle Scholar
  27. 27.
    Zäihle, M., Ziezold, H.: Fractional derivatives of Weierstrass-type functions. J. Comput. Appl. Math. 76, 265–275 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Faculty of SciencesAlexandria UniversityAlexandriaEgypt

Personalised recommendations