Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

ON CONVERGENCE THEOREMS FOR THE MCSHANE INTEGRAL ON TIME SCALES

  • You, Xuexiao (School of Mathematics and Statistics Hubei Normal University) ;
  • Zhao, Dafang (School of Mathematics and Statistics Hubei Normal University)
  • Published : 2012.08.15
⟨ Previous Next ⟩

Abstract

In this paper, we study the process of McShane delta integrals on time scales and discuss the relation between McShane delta integral and Henstock delta integral. We also prove the mono- tone convergence theorem, Fatou's Lemma and the dominated con- vergence theorems for the McShane delta integral.

Keywords

References

  1. S. Hilger, Ein Makettenkalkl mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph. D. Thesis, Universtat Wurzburg, 1988.
  2. S. Hilger, Analysis on measure chainsA unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18-56. https://doi.org/10.1007/BF03323153
  3. M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2004.
  4. A. Peterson, B. Thompson, HenstockCKurzweil Delta and Nabla Integrals, J. Math. Anal. Appl. 323 (2006), 162-178. https://doi.org/10.1016/j.jmaa.2005.10.025
  5. G. Sh. Guseinov, Integration on time scales, J. Math. Anal. Appl. 285 (2003), 107-127. https://doi.org/10.1016/S0022-247X(03)00361-5
  6. G. Sh. Guseinov, B. Kaymakcalan, Basics of Riemann delta and nabla integra- tion on time scales, J. Difference Equations Appl. 8 (2002), 1001-1027. https://doi.org/10.1080/10236190290015272
  7. S. Avsec, B. Bannish, B. Johnson, and S. Meckler, The Henstock-Kurzweil delta integral on unbounded time scales, PanAmerican Math. J. 16 (2006), no. 3, 77-98.
  8. B. S. Thomson, Henstock-Kurzweil integrals on time scales, PanAmerican Math. J. 18 (2008), no. 1, 1-19.
  9. L. P. Yee and R. Vyborny, The integral, An Easy Approach after Kurzweil and Henstock, Australian Mathematical Society Lecture Series 14, Cambridge University Press, 2000.
  10. C. W. Swartz, Douglas S Kurtz, Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and Mcshane, World Scientific, 2004.
  11. D. Zhao and G. Ye, C-integral and Denjoy-C integral, Comm. Korean. Math. Soc. 22 (2007), no. 1, 27-39. https://doi.org/10.4134/CKMS.2007.22.1.027
  12. D. Zhao and G. Ye, On AP-Henstock-Stieltjes integral, J. Chungcheong Math. Soc. 19 (2006), no. 2, 177-188.
  13. D. Zhao and G. Ye, On strong C-integral of Banach-valued functions, J. Chungcheong Math. Soc. 20 (2007), no. 1, 1-10.
  14. J. M. Park, The Denjoy extension of the McShane integral, Bull. Korean Math. Soc. 33 (1996), no. 3, 411-416.

Cited by

  1. ON DELTA ALPHA DERIVATIVE ON TIME SCALES vol.29, pp.2, 2016, https://doi.org/10.14403/jcms.2016.29.2.255
  2. Some inequalities for interval-valued functions on time scales pp.1433-7479, 2019, https://doi.org/10.1007/s00500-018-3538-6