Generalized Implicit Function Theorem and General Fundamental Theorem of Calculus
Abstract
We present the notion of Henstock-Kurzweil integral for mappings assuming values in Hausdorff topological vector spaces using the direct set of gauges and derive a version of Mean Value Theorem. We use the definition of Frechet derivative and obtain a general version of Implicit Function Theorem for mappings from X \times Y \rightarrow Z where, for existence and continuity of the function, X needs to be merely a topological space and for differentiability, X can be a Topological Vector Space (TVS) while Z is a Hausdorff topological vector space and Y is a Banach space. The implicit function theorem is proved in 3 parts as existence, continuity of the partial derivative and invertibility of the partial derivative. The proof is very similar to the classical proof.
Full text article
References
J. Dieuodenne, Foundations of Modern Analysis. Academic Press, 1956. https://www. perlego.com/book/3457542/foundations-of-modern-analysis-pdf.
A. Willansky, Functional Analysis. Blaisdell Publishing Company, 1964.
J. Lloyd, “Two topics in the differential calculus on topological linear spaces,” Bulletin of the Australian Mathematical Society, vol. 9, no. 2, pp. 305–306, 1973.
H. Kalita and B. Hazarika, “A convergence theorem for ap-henstock-kurzweil integral and its relation to topology,” Filomat, vol. 36, no. 20, pp. 6831–6839, 2022. https://doi.org/10.2298/FIL2220831K.
R. Bartle, A modern theory of Integrations. American mathematical Society, 2001. https://bookstore.ams.org/view?ProductCode=GSM/32.
A. Boccuto and B. Riečan, “On the henstock-kurzweil integral for riesz-space-valued functions defined on unbounded intervals,” Czechoslovak Mathematical Journal, vol. 54, pp. 591––607, 2004. https://doi.org/10.1007/s10587-004-6411-y.
R. J. Aumann, “Integrals of set valued functions,” Journal Of Mathematical Analysis and Applications, vol. 12, no. 1, pp. 1–12, 1965. https://www.sciencedirect.com/journal/journal-of-mathematical-analysis-and-applications/vol/12/issue/1.
T. G. Thange and S. S. Gangane, “Henstock - kurzweil integral for banach valued function,” Mathematics and Statistics, vol. 10, no. 5, pp. 1038–1049, 2022. DOI:10.13189/ms.2022.100515.
A. Pedgaonkar, Henstock-Kurzeweil Integration. Ph. D. thesis submitted to BAMU, Aurangabad, 2014.
S. Heikkila, “Differential and integral equations with henstock–kurzweil integrable functions,” Journal of Mathematical Analysis and Applications, vol. 379, no. 1, pp. 171–179, 2011. doi:10.1016/j.jmaa.2010.12.050.
A. Pedgaonkar, “Fundamental theorem of calculus for henstock-kurzeweil integral,” Bulletin of Marathwada Mathematical Society, vol. 14, no. 1, pp. 71–80, 2013.
S. Kesavan, Non-linear Functional Analysis:A First Course. Hindusthan Book Agency, 2004. http://www.hindbook.com/index.php/nonlinear-functional-analysis-a-first-course.
D. Piazza and K. Musial, “Set-valued kurzweil–henstock–pettis integral,” Set-Valued Analysis, vol. 13, pp. 167—-179, 2005. https://doi.org/10.1007/s11228-004-0934-0.
X. You, D. Zhao, and D. F. M. Torres, “On the henstock-kurzweil integral for riesz-space-valued functions on time scales,” J. Nonlinear Sci. Appl., vol. 10, no. 5, pp. 2487––2500, 2017. http://dx.doi.org/10.22436/jnsa.010.05.18.
S. K. Gudade, S. K.Panchal, and D. B. Dhaigude, “Existence results for weakly coupled systems of ψ-caputo fractional differential equations with nonlinear boundary conditions,” J. Math. Comput. Sci., vol. 12, pp. 1–15, 2022. https://scik.org/index.php/jmcs/article/view/6805.
T. Apostol, Mathematical Analysis. Narosa Publising House, 2002. http://www.narosa.com/books_display.asp?catgcode=978-81-85015-66-8.
S. Lang, Analysis I. Addison Wesley, 1956.
Authors
Copyright (c) 2026 Journal of the Indonesian Mathematical Society
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Article Details
