Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

RIEMANN-LIOUVILLE FRACTIONAL FUNDAMENTAL THEOREM OF CALCULUS AND RIEMANN-LIOUVILLE FRACTIONAL POLYA TYPE INTEGRAL INEQUALITY AND ITS EXTENSION TO CHOQUET INTEGRAL SETTING

  • Received : 2018.10.02
  • Accepted : 2019.08.14
  • Published : 2019.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

Here we present the right and left Riemann-Liouville fractional fundamental theorems of fractional calculus without any initial conditions for the first time. Then we establish a Riemann-Liouville fractional Polya type integral inequality with the help of generalised right and left Riemann-Liouville fractional derivatives. The amazing fact here is that we do not need any boundary conditions as the classical Polya integral inequality requires. We extend our Polya inequality to Choquet integral setting.

Keywords

References

  1. G. A. Anastassiou, Intelligent mathematics: computational analysis, Intelligent Systems Reference Library, 5, Springer-Verlag, Berlin, 2011. https://doi.org/10.1007/978-3-642-17098-0
  2. G. Choquet, Theory of capacities, Ann. Inst. Fourier, Grenoble 5 (1953), 131-295. https://doi.org/10.5802/aif.53
  3. D. Denneberg, Non-additive measure and integral, Theory and Decision Library. Series B: Mathematical and Statistical Methods, 27, Kluwer Academic Publishers Group, Dordrecht, 1994. https://doi.org/10.1007/978-94-017-2434-0
  4. I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.
  5. G. Polya, Ein Mittelwertsatz fur Funktionen mehrerer Veranderlichen, Tohoku Math. J. 19 (1921), 1-3. https://doi.org/10.2748/tmj/1178243344
  6. G. Polya and G. Szego, Aufgaben und Lehrsatze aus der Analysis. Band I, Dritte berichtigte Au age. Die Grundlehren der Mathematischen Wissenschaften, Band 19, Springer-Verlag, Berlin, 1964.
  7. G. Polya and G. Szego, Problems and theorems in analysis. Vol. I, translated from the German by D. Aeppli, Springer-Verlag, New York, 1972.
  8. G. Polya and G. Szego, Problems and theorems in analysis. Vol. II, revised and enlarged translation by C. E. Billigheimer of the fourth German edition, springer Study Edition, Springer-Verlag, New York, 1976.
  9. F. Qi, Polya type integral inequalities: origin, variants, proofs, refinements, generalizations, equivalences, and applications, Math. Inequal. Appl. 18 (2015), no. 1, 1-38. https://doi.org/10.7153/mia-18-01
  10. M. Sugeno, A note on derivatives of functions with respect to fuzzy measures, Fuzzy Sets and Systems 222 (2013), 1-17. https://doi.org/10.1016/j.fss.2012.11.003
  11. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, reprint of the fourth (1927) edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. https://doi.org/10.1017/CBO9780511608759