Conformable Fractional Approximation by Choquet Integrals

  • George A. AnastassiouEmail author
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 190)


Here we present the conformable fractional quantitative approximation of positive sublinear operators to the unit operator. These are given a precise Choquet integral interpretation. Initially we start with the study of the conformable fractional rate of the convergence of the well-known Bernstein–Kantorovich–Choquet and Bernstein–Durrweyer–Choquet polynomial Choquet-integral operators. Then we study in the fractional sense the very general comonotonic positive sublinear operators based on the representation theorem of Schmeidler [12]. We continue with the conformable fractional approximation by the very general direct Choquet-integral form positive sublinear operators. The case of convexity is also studied thoroughly and the estimates become much simpler. All approximations are given via inequalities involving the modulus of continuity of the approximated function’s higher order conformable fractional derivative. It follows [5].


  1. 1.
    M. Abu Hammad, R. Khalil, Abel’s formula and Wronskian for conformable fractional differential equations. Int. J. Differ. Equ. Appl. 13(3), 177–183 (2014)Google Scholar
  2. 2.
    G. Anastassiou, Moments in Probability and Approximation Theory. Pitman Research Notes in Mathematics Series (Longman Group UK, New York, 1993)Google Scholar
  3. 3.
    G. Anastassiou, Conformable fractional approximation by max-product operators. Studia Mathematica Babes Bolyai 63(1), 3–22 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    G. Anastassiou, Conformable fractional approximations by max-product operators using convexity. Arab. J. Math. (2018). Accepted for publicationGoogle Scholar
  5. 5.
    G.A. Anastassiou, Conformable fractional approximation by Choquet integrals. J. Comput. Anal. Appl. (2018). AcceptedGoogle Scholar
  6. 6.
    G. Choquet, Theory of capacities. Ann. Inst. Fourier (Grenoble) 5, 131–295 (1954)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D. Denneberg, Non-additive Measure and Integral (Kluwer, Dordrecht, 1994)CrossRefGoogle Scholar
  8. 8.
    D. Dubois, H. Prade, Possibility Theory (Plenum Press, New York, 1988)CrossRefGoogle Scholar
  9. 9.
    S. Gal, Uniform and pointwise quantitative approximation by Kantorovich–Choquet type integral operators with respect to monotone and submodular set functions. Mediterr. J. Math. 14(5), 12 pp. (2017). Art. 205Google Scholar
  10. 10.
    S. Gal, S. Trifa, Quantitative estimates in uniform and pointwise approximation by Bernstein–Durrmeyer–Choquet operators. Carpathian J. Math. 33(1), 49–58 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    D. Schmeidler, Integral representation without additivity. Proc. Am. Math. Soc. 97, 255–261 (1986)MathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Schmeidler, Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)MathSciNetCrossRefGoogle Scholar
  14. 14.
    L.S. Shapley, A Value for n-person games, in Contributions to the Theory of Games, ed. by H.W. Kuhn, A.W. Tucker. Annals of Mathematical Studies, vol. 28 (Princeton University Press, Princeton, 1953), pp. 307–317Google Scholar
  15. 15.
    Z. Wang, G.J. Klir, Generalized Measure Theory (Springer, New York, 2009)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

Personalised recommendations