A Review of Pompeiu-Hausdorff Metric Differentiability and Its Relation to Generalized Hukuhara Differentiability

William Surya Putra, Mohamad Muslikh

Abstract


The Pompeiu-Hausdorff distance/Pompeiu-Hausdorff metric is a concept in analysis that measures the distance between two subsets of a metric space, one of its important applications being the Hausdorff metric differentiability of set-valued functions . This article reviews the definition and properties of Pompeiu-Hausdorff distance differentiability on the space of compact and convex subsets of the Euclidean space with dimension n. We also present concepts about the generalized Hukuhara difference and its differentiability. By studying both topics, we discuss the established relationship between Pompeiu-Hausdorff metric differentiability and generalized Hukuhara differentiability

Keywords


matematika;analisis;turunan metrik;hukuhara

Full Text:

PDF

References


[1] Aubin and Frankowska. Set-Valued Analysis. Vol. 1. Birkhäuser Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0.

[2] Aubin and Frankowska. “Introduction: Set-Valued Analysis in Control Theory”. In: Set-Valued Analysis 8.1 (2000), pp. 1–9. doi: 10.1023/A:1008724221942.

[3] D. P. Huttenlocher, G. A. Klanderman, and W. J. Rucklidge. “Comparing images using the Hausdorff distance”. In: IEEE Transactions on Pattern Analysis and Machine Intelligence 15.9 (1993), pp. 850–863. doi: 10.1109/34.232073.

[4] O.H. Huseynov R.A. Aliev Witold Pedrycz. “Hukuhara difference of Z-numbers”. In: Information Sciences 466.2 (2018), pp. 13–24. doi: 10.1016/j.ins.2018.07.033.

[5] Y. Chalco-Cano et al. “Calculus for interval-valued functions using generalized Hukuhara derivative and applications”. In: Fuzzy Sets and Systems 219 (2013), pp. 49–67. doi: 10.1016/j.fss.2012.12.004.

[6] M. Muslikh et al. “The metric derivative of set-valued functions”. In: Advances in Pure and Applied Mathematics 10.3 (2019), pp. 263–272. doi: 10.1515/apam-2018-0028.

[7] Masuo Hukuhara. “Integration des applications mesurables dont la valeur est un compact convex”. In: Funkcialaj Ekvacioj 18.10 (1967), pp. 205–223.

[8] Barnabás Bede Luciano Stefanini. “Generalized Hukuhara differentiability of interval-valued functions and interval differential equations”. In: Nonlinear Analysis: Theory, Methods & Applications 71.10 (2009), pp. 1311–1328. doi: 10.1016/j.na.2008.12.005.

[9] M. Shahidia A. Khastana R. Rodríguez-Lópezb. “New differentiability concepts for set-valued functions and applications to set differential equations”. In: Information Sciences 575.21 (2021), pp. 355–378. doi: 10.1016/j.ins.2021.06.014.

[10] Fitri S. Muslikh M. Fungsi Bernilai Himpunan. Vol. 1. UBPress, 2022.

[11] Roger J. B. Wets R. Tyrrell Rockafellar. Variational Analysis. Vol. 317. Springer Berlin, Heidelberg, 1998. doi: 10.1007/978-3-642-02431-3.

[12] Rafal K. Goebel. Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control: An Introduction. Vol. 197. SIAM (Society for Industrial and Applied Mathematics), 2024. doi: 10.1137/1.9781611977981.

[13] Arrigo C. Aubin. Differential Inclusions. Springer Berlin, Heidelberg, 1984. doi: 10.1007/978-3-642-69512-4.

[14] Schneider R. Convex Bodies: The Brunn–Minkowski Theory. Vol. 51. Cambridge University Press, 2013. doi: 10.1017/CBO9781139003858.

[15] Valeri O. Aram V. Convex and Set-Valued Analysis. Vol. 1. Deutsche Nationalbibliothek, 2016. doi: 10.1515/9783110460308-030.

[16] A. Sheldon. Linear Algebra Done Right. Vol. 4. Springer Cham, 2023. doi: 10.1007/978-3-031-41026-0.

[17] Mahlon M. Day. Normed Linear Spaces. Vol. 21. Springer Berlin, Heidelberg, 1958. doi: 10.1007/978-3-662-25249-9.

[18] Sen R. A First Course in Functional Analysis: Theory and Applications. Vol. 21. Anthem Press, 2013. doi: 10.7135/9780857282224.

[19] J.R. Giles. Introduction to the Analysis of Normed Linear Spaces. Cambridge University Press, 2012. doi: 10.1017/CBO9781139168465.

[20] Claude Lemaréchal Jean-Baptiste Hiriart-Urruty. Fundamentals of Convex Analysis. Vol. 305. Springer Berlin, Heidelberg, 2001. doi: 10.1007/978-3-642-56468-0.

[21] Lieven Vandenberghe Stephen Boyd. Convex Optimization. Cambridge University Press, 2004. doi: 10.1017/CBO9780511804441.

[22] A. Barvinok. A Course in Convexity. Vol. 54. American Mathematical Society, 2002. https://bookstore.ams.org/gsm-54/#.

[23] Stanislaw H. Żak Edwin K. P. Chong. An Introduction to Optimization. John Wiley & Sons, 2008. doi: 10.1002/9781118033340.

[24] E.T. Copson. Metric Spaces. Cambridge: Cambridge University Press, 2010. doi: 10.1017/CBO9780511566141.

[25] W. Rucklidge. The Hausdorff distance. Vol. 1173. Springer, Berlin, Heidelberg, 1996. doi: 10.1007/BFb0015093.

[26] Kloeden P. Diamond P. Metric Spaces of Fuzzy Sets: Theory and Applications. Vol. 1. World Scientific, 1994. doi: 10.1142/2326.

[27] L Stefanini. “A Generalization of Hukuhara Difference”. In: Advances in Soft Computing 48 (2008), pp. 203–210. doi: 10.1007/978-3-540-85027-4_25.

[28] S. Markov. “Calculus for interval functions of a real variable”. In: Computing 22 (1979), pp. 325–337. doi: 10.1007/BF02265313.




DOI: https://doi.org/10.18860/cauchy.v11i1.37820

Refbacks

  • There are currently no refbacks.


Copyright (c) 2026 William Surya Putra

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Editorial Office
Mathematics Department,
Maulana Malik Ibrahim State Islamic University of Malang
Gajayana Street 50 Malang, East Java, Indonesia 65144
e-mail: cauchy@uin-malang.ac.id

Creative Commons License
CAUCHY: Jurnal Matematika Murni dan Aplikasi is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.