Sufficient Conditions for Convergence of Sequences of Henstock-Kurzweil Integrable Functions

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DOI: https://doi.org/10.28924/2291-8639-21-2023-49
Published: Jun 2, 2023
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Yassin Alzubaidi, Sufficient Conditions for Convergence of Sequences of Henstock-Kurzweil Integrable Functions, Int. J. Anal. Appl., 21 (2023), 49.

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Yassin Alzubaidi

Abstract

The main aim of this paper is to present our approach of obtaining sufficient conditions for convergence of sequences of Henstock-Kurzweil integrable functions. Our approach involves the use of the concept of multiplier functions, where we define a class Φ of multipliers for the Henstock-Kurzweil integral. We consider a sequence (fn) of Henstock-Kurzweil integrable functions on a non-degenerate interval [a, b] and we assume that (fn) converges point wise to a function f. Then we show that f is Henstock-Kurzweil integrable and its integral is equal to the limit of the sequence (∫abfn) if there exists φ∈Φ such that the defined functionals of the type F (φ, fn) satisfy the imposed conditions. Beside the fact that the results regarding the convergence under the integral sign are always of great importance, the method introduced here can be imitated and used to obtain other results on the related areas.

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References

  1. Y. Alzubaidi, A Complete Normed Space of a Class of Guage Integrable Functions, J. Math. 2022 (2022), 2022/2354758. https://doi.org/10.1155/2022/2354758.
  2. R.G. Bartle, Return to the Riemann Integral, Amer. Math. Mon. 103 (1996), 625–632. https://doi.org/10.1080/00029890.1996.12004798.
  3. J. Depree, C. Swartz, Introduction to Real Analysis, Wily, New York, 1987.
  4. R.A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, American Mathematical Society, Providence, R.I, 1994.
  5. R. Henstock, Definitions of Riemann Type of the Variational Integrals, Proc. London Math. Soc. s3-11 (1961), 402–418. https://doi.org/10.1112/plms/s3-11.1.402.
  6. J. Kurzweil, Generalized Ordinary Differential Equations and Continuous Dependence on a Parameter, Czechoslovak Math. J. 82 (1957), 418-446.
  7. H.L. Royden, P.M. Fitzpatrick, Real Analysis, 4th Edition, Pearson Education, 2010.
  8. C. Swartz, Norm convergence and uniform integrability for the Henstock-Kurzweil Integral, Real Anal. Exchange, 24 (1998/99), 423-426.
  9. C. Swartz, Introduction to Gauge Integrals, World Scientific, Singapore, 2001.
  10. W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, New York, 1976.
  11. T.Y. Lee, Henstock-Kurzweil integration on Euclidean spaces, World Scientific, Singapore, 2011.