A Strong Left Fractional Calculus for Banach Space Valued Functions

  • George A. AnastassiouEmail author
Part of the Studies in Computational Intelligence book series (SCI, volume 734)


We develop here a strong left fractional calculus theory for Banach space valued functions of Caputo type. Then we establish many Bochner integral inequalities of various types. This chapter is based on Anastassiou (A strong fractional calculus theory for banach space valued functions, 2017 [5]).


  1. 1.
    R.P. Agarwal, V. 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
  2. 2.
    C.D. Aliprantis, K.C. Border, Infinite Dimensional Analysis (Springer, New York, 2006)zbMATHGoogle Scholar
  3. 3.
    G. Anastassiou, Fractional Differentiation Inequalities (Springer, New York, 2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    G. Anastassiou, Advances on fractional inequalities (Springer, New York, 2011)CrossRefzbMATHGoogle Scholar
  5. 5.
    G. Anastassiou, A Strong Fractional Calculus Theory for Banach Space Valued Functions, Nonlinear Functional Analysis and Applications (2017) (accepted)Google Scholar
  6. 6.
    F. Appendix, The Bochner Integral and Vector-Valued \( L_{p}\) -Spaces,
  7. 7.
  8. 8.
    K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, 2004 (Springer, Berlin, 2010)Google Scholar
  9. 9.
    V. Kadets, B. Shumyatskiy, R. Shvidkoy, L. Tseytlin, K. Zheltukhin, Some remarks on vector-valued integration. Math. Fiz. Anal. Geom. 9(1), 48–65 (2002)MathSciNetzbMATHGoogle Scholar
  10. 10.
    M. Kreuter, Sobolev Spaces of Vector-Valued Functions, Ulm University, Master Thesis in Mathematics, Ulm, Germany, 2015Google Scholar
  11. 11.
    E. Landau, Einige Ungleichungen für zweimal differentzierban funktionen. Proc. London Math. Soc. 13, 43–49 (1913)zbMATHGoogle Scholar
  12. 12.
    J. Mikusinski, The Bochner Integral (Academic Press, New York, 1978)CrossRefzbMATHGoogle Scholar
  13. 13.
    G.E. Shilov, Elementary Functional Analysis (Dover Publications Inc., New York, 1996)zbMATHGoogle Scholar
  14. 14.
    C. Volintiru, A proof of the fundamental theorem of Calculus using Hausdorff measures. Real Analysis Exchange, 26(1), 381–390 (2000/2001)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

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