Hadamard-type fractional calculus in Banach spaces

  • Hussein A. H. Salem
Original Paper


In this pages, we present the definitions and some properties of the Hadamard-type fractional Pettis integrals (and corresponding fractional derivatives) for the functions that take values in Banach space. Further, we show that a well known properties of the Hadamard-type fractional calculus over the space of real-valued functions also hold in Banach spaces. Some emphasizes examples are demonstrated. Meanwhile, we construct an example of a function that has no pseudo derivative everywhere, but has a Hadamard-type fractional derivative. As far as we know, the topic of this paper was never investigated before, and so is new.


Fractional calculus Pettis integrals 

2000 Mathematics Subject Classification

26A33 34G20 



I would like to express my gratitude to Prof. Martin Väth, for his advise, guidance, patience and continuous support.


  1. 1.
    Agarwal, R.P. Lupulescu, V., ORegan, D., Rahman, G.: Multi-term fractional differential equations in a nonreflexive Banach space. Adv. Diff. Equ, 2013, 302 (2013)Google Scholar
  2. 2.
    Agarwal, R.P., Vasile Lupulescu, D., O’Regan, G.Rahman: Fractional calculus and fractional differential equations in nonreflexive Banach spaces. Commun. Nonlinear Sci. Numer. Simulat. 20, 59–73 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Agarwal, R.P., Lupulescu, V., O’Regan, D., Rahman, G.: Nonlinear fractional differential equations in nonreflexive Banach spaces and fractional calculus. Adv. Diff. Equ. 2015, 112 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ahmad, B., Ntouyas, S.K.: A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations. Fract. Calc. Appl. Anal. 17(2), 348–360 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Alexiewicz, A.: On the differentiation of vector-valued functions. Stud. Math. 11(1), 185–196 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ball, J.M.: Weak continuity properties of mapping and semi-groups, Proc. Royal Soc. Edinbourgh Sect. A 72, 275–280 (1973–1974)Google Scholar
  7. 7.
    Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 270(1), 1–15 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 269(2), 387–400 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Corduneanu, C.: Integral Equations and Applications. Cambridge University Press, New York (1991)CrossRefzbMATHGoogle Scholar
  10. 10.
    Diestel, J., Uhl Jr., J.J.: Vector Measures, Math. Surveys 15. Amer. Math. Soc, Providence (1977)CrossRefzbMATHGoogle Scholar
  11. 11.
    Edgar, G.A.: Measurability in a Banach spaces. Indiana Univ. Math. J. 26(6), 663–677 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Edgar, G.A.: Measurability in a Banach spaces, II. Indiana Univ. Math. J. 28(4), 559–578 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Floret, K.: Weakly Compact Sets, Lecture Notes in Math, vol. 801. Springer, Berlin (1980)CrossRefGoogle Scholar
  14. 14.
    Gàmaz, J., Mendoza, J.: On Denjoy-Dunford and Denjoy-Pettis integrals. Studia Math. 130(2), 155–133 (1998)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gordon, R.: The Denjoy extension of Bochnar, Pettis and Dunford integrals, Studia Math. T. XCII. 73–91 (1992–1993)Google Scholar
  16. 16.
    Geitz, R.F.: Pettis integration. Proc. Am. Math. Soc. 82, 81–86 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Geitz, R.F.: Geomerty and the Pettis integral. Trans. Am. Math. Soc. 269, 535–548 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hille, E., Phillips, R.S.: Functional analysis and semi-groups, vol. 31. Amer. Math. Soc. Colloq., Providence (1957)zbMATHGoogle Scholar
  19. 19.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science Limited, Amsterdam, the Netherlands (2006)Google Scholar
  20. 20.
    Kilbas, A.A.: Hadamard-type fractional calculus. J. Kor. Math. Soc. 38(6), 1191–1204 (2001)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Klimek, M.: Sequential fractional differential equations with Hadamard derivative. Commun. Nonlinear Sci. Numer. Simul. 16(12), 4689–4697 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Knight, W.J.: Absolute continuity of some vector functions and measures. Can. J. Math. 24(5), 73746 (1972)MathSciNetGoogle Scholar
  23. 23.
    Martin, R.H., Uhl Jr., : Nonlinear Operators and Differential Equations in Banach Spaces. Wiley, New York (1976)Google Scholar
  24. 24.
    Mitchell, A.R., Smith, Ch.: An existence theorem for weak solutions of differential equations in Banach spaces. In: Lakshmikantham, V. (ed.) Nonlinear Equations in Abstract Spaces, pp. 387–404 (1978)Google Scholar
  25. 25.
    Naralenkov, K.: On Denjoy type extension of the Pettis integral. Czechoslovak. Math. J. 60(135), 737–750 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    O’Regan, D.: Integral equations in reflexive Banach spaces and weak topologies. Proc. Am. Math. Soc. 124(2), 607–614 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pettis, B.J.: On integration in vector spaces. Trans. Am. Math. Soc. 44, 277–304 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Salem, H.A.H., El-Sayed, A.M.A.: Weak solution for fractional order integral equations in reflexive Banach spaces. Mathematica Slovaca 55(2), 169–181 (2005)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Salem, H.A.H., El-Sayed, A.M.A., Moustafa, O.L.: A note on the fractional calculus in Banach spaces. Studia Sci. Math. Hungar. 42(2), 115–130 (2005)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Salem, H.A.H.: On the nonlinear Hammerstein integral equations in Banach spaces and application to the boundary value problem of fractional order. Math. Comput. Modell. 48, 1178–1190 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Salem, H.A.H.: Multi-Term fractional differential equation in reflexive Banach spaces. Math. Comput. Modell. 49, 829–834 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    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
  33. 33.
    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
  34. 34.
    Salem, H.A.H.: Quadratic integral equations in reflexive Banach spaces. Discuss. Math. Differ. Incl. Control Optim. 30, 61–69 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Salem, H.A.H.: On the quadratic integral equations and their applications. Comput. Math. Appl. 62(8), 2931–2943 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Salem, H.A.H., Cichoń, M.: On solutions of fractional order boundary value problems with integral boundary conditions in Banach spaces. J. Funct. Sp. Appl. 428094, 13 (2015)zbMATHGoogle Scholar
  37. 37.
    Samko, S., Kilbas, A., Marichev, O.L.: Fractional Integrals and Drivatives. Gordon and Breach Science Publishers, Longhorne, PA, (1993)Google Scholar
  38. 38.
    Schwabik, S., Guoju, Y.: Topics in Banach space integration. World Scientific, Singapore (2005)CrossRefzbMATHGoogle Scholar
  39. 39.
    Solomon, D.: On differentiability of vector-valued functions of a real variables. Stud. Math. 29, 1–4 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Solomon, D.: Denjoy Integration in Abstract Spaces, Memories of the American Mathematical Society. American Mathematical Society, Providence (1969)Google Scholar
  41. 41.
    Szep, A.: Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces. Stud. Sci. Math. Hungar. 6, 197–203 (1971)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Szulfa, S.: Sets of fixed points of nonlinear mappings in function spaces. Funkial Ekvac. 22, 121–126 (1979)MathSciNetGoogle Scholar
  43. 43.
    Zhang, X., Liu, Z., Peng, H., Zhang, X., Yang, S.: The general solution of differential equations with Caputo-Hadamard fractional derivatives and noninstantaneous impulses. Adv. Math. Phys. 3094173, 11 (2017)MathSciNetGoogle Scholar

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© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

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

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