Equivalence of sequential Henstock and topological Henstock integrals for interval valued functions

Document Type : Research Paper

Authors

Department of Mathematics, University of Lagos, Lagos, Nigeria

Abstract

Suppose $X$ is a locally compact Hausdorff space and $\Omega \in \bigtriangleup$. If $ F $ is an interval valued function defined in $ \Omega $ with $F:\bar \Omega\rightarrow I_{\mathbb{R}}$. Suppose $F$ is Topological Henstock integrable, is $ F $ Sequential Henstock integrable? Therefore, the purpose of this paper is to provide a positive response to this query.

Keywords


[1] J. A. Chart eld, Equivalence Of Integrals, Proceedings of American Mathematical Society. in Mathematics. Vol. 3, (1973), 279-285.
[2] O. Holzmann, B., Lang, and H. Schutt, H. Newton's constant of gravitation and veri ednumerical quadratures., Reliable Compute., 2(3), (1996), 229-240.
[3] M. E. Hamid, L. Xu, and Z. Gong, The Henstock-Stieltjes Integral For Set Valued Functions., International Journal of Pure and Applied Mathematics. Volume 114. No. 2. (2017), 261-275.
[4] M. E. Hamid, A. H. and Elmuiz, On Henstock-Stieltjes Integrals of interval-Valued Functions and Fuzzy-Number-Valued Functions, Journal of Applied Mathematics and Physics, 4. (2016), 779-786.
[5] V. O. Iluebe, and A. A. Mogbademu, . Equivalence Of Henstock And Certain Sequential Henstock Integral, Bangmond International Journal of Mathematics and Computational Science. Vol. 1. No. 1 and 2, (2020), 9 - 16.
[6] V. O. Iluebe, and A. A. Mogbademu, Sequential Henstock Integral For Interval Valued Functions, . CJMS. 11(2022), 358-367.
[7] V. O. Iluebe, and A. A. Mogbademu, Equivalence Of P-Henstock Type , Annal of Mathematics and Computational Science. Vol. 2. (2021), 15-22.
[8] V. O. Iluebe, and A. A. Mogbademu, On Sequential Henstock Stieltjes Integral For Lp[0; 1]-Interval Valued Functions, Bull. Int. Math. Virtual Inst., 12(2022), 369 - 378.
[9] W. Kramer, and S. Wedner, Two adaptive Gauss-Legendre type algorithms for the veri ed computation of de nite integrals, Reliable Compute., 2(3),(1996), 241-254.
[10] Y. J. Kim, The Equivalence of Perron, Henstock and Variational Stieltjes Integrals, Journal of the Chungcheong Mathematics Society, Vol. 10, (1997), 29-36.
[11] B. Lang, Derivative-based subdivision in multi-dimensional veri ed Gaussian quadrature., in Alefeld, G., Rohn, J., Siegfried M. Rump, and Yamamoto, T. Symbolic Algebraic Methods and Veri cation Methods, Springer-Verlag, New York, 2001, pp. 145-152.
[12] R. E. Moore, R. B.Kearfott, R. B. and J. C. Michael, Introduction to Interval Analysis., Society for Industrial and Applied Mathematics.(2009), 37-38; 129-135.
[13] L. A. Paxton, Sequential Approach to the Henstock Integral, .Washington State University, arXiv:1609.05454v1 [maths.CA] 18 Sep, (2016), 3-5.
[14] M. C. Ray, Equivalence Of Riemann Integral Based on p-Norm, School Of Mathematics and Statistics. Vol.6, (2008), 1-13.
[15] L. Solomon, The Kursweil-Henstock Integral and Its Applications, . Marcel Dekker, (2001), 11-25.
[16] C. X. Wu, and Z. T. Gong, On Henstock Integrals of interval-Valued Functions and Fuzzy-Number-Valued Functions, Fuzzy Set and Systems. 115, (2016), 377-391.
17] J. L. Ying, On The Equivalence Of Mcshane and Lebesgue Integrals, Real Analysis Exchange. Vol. 21(2), (1995-96), 767-770.