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

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References


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

[2] Aubin and Frankowska, “Introduction: Set-valued analysis in control theory,” Set-Valued Analysis, vol. 8, no. 1, pp. 1–9, 2000. doi: https://doi.org/10.1023/A:1008724221942.

[3] D. P. Huttenlocher, G. A. Klanderman, and W. J. Rucklidge, “Comparing images using the hausdorff distance,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 15, no. 9, pp. 850–863, 1993. doi: https://doi.org/10.1109/34.232073.

[4] O.H.R.A.Aliev Witold Pedrycz, “Hukuhara difference of z-numbers,” Information Sciences, vol. 466, no. 2, pp. 13–24, 2018. doi: https://doi.org/10.1016/j.ins.2018.07.033.

[5] Y. Chalco-Cano, A. Rufián-Lizana, H. Román-Flores, and M. Jiménez Gamero, “Calculus for interval-valued functions using generalized hukuhara derivative and applications,” Fuzzy Sets and Systems, vol. 219, pp. 49–67, 2013. doi: 10.1016/j.fss.2012.12.004. William Surya Putra,Mohamad Muslikh 438 316 A Review of Pompeiu-Hausdorff Metric Differentiability and Its Relation to Generalized Hukuhara Differentiability

[6] M. Muslikh, A. Kilicman, S. H. Bt Sapar, and N. Bt Bachoklati, “The metric derivative of set-valued functions,” Advances in Pure and Applied Mathematics, vol. 10, no. 3, pp. 263 272, 2019. doi: 10.1515/apam-2018-0028.

[7] M. Hukuhara, “Integration des applications mesurables dont la valeur est un compact convex,” Funkcialaj Ekvacioj, vol. 18, no. 10, pp. 205–223, 1967.

[8] B. B. Luciano Stefanini, “Generalized hukuhara differentiability of interval-valued functions and interval differential equations,” Nonlinear Analysis: Theory, Methods Applications, vol. 71, no. 10, pp. 1311–1328, 2009. doi: https://doi.org/10.1016/j.na.2008.12.005.

[9] M. S. A. Khastana R. Rodríguez-Lópezb, “New differentiability concepts for set-valued functions and applications to set differential equations,” Information Sciences, vol. 575, no. 21, pp. 355–378, 2021. doi: https://doi.org/10.1016/j.ins.2021.06.014.

[10] F. S. Muslikh M., FUNGSI BERNILAI HIMPUNAN. UBPress, 2022, vol. 1.

[11] R. J. B. W. R. Tyrrell Rockafellar, Variational Analysis. Springer Berlin, Heidelberg, 1998, vol. 317. doi: https://doi.org/10.1007/978-3-642-02431-3.

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

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

[14] S. R., Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press, 2013, vol. 51. doi: https://doi.org/10.1017/CBO9781139003858.

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

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

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

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

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

[20] C. L. Jean-Baptiste Hiriart-Urruty, Fundamentals of Convex Analysis. Springer Berlin, Heidelberg, 2001, vol. 305. doi: https://doi.org/10.1007/978-3-642-56468-0.

[21] L. V. Stephen Boyd, Convex Optimization. Cambridge University Press, 2004. doi: https: //doi.org/10.1017/CBO9780511804441.

[22] A. Barvinok, A Course in Convexity. American Mathematical Society, 2002, vol. 54. Available online.

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

[24] E. Copson, Metric Spaces. Cambridge: Cambridge University Press, 2010. doi: https://d oi.org/10.1017/CBO9780511566141.

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

[26] K. P. Diamond P., Metric Spaces of Fuzzy Sets: Theory and Applications. World Scientific, 1994, vol. 1. doi: https://doi.org/10.1142/2326. William Surya Putra,Mohamad Muslikh 360 439 361 A Review of Pompeiu-Hausdorff Metric Differentiability and Its Relation to Generalized Hukuhara Differentiability

[27] L. Stefanini, “A generalization of hukuhara difference,” Advances in Soft Computing, vol. 48, 362 363 364 pp. 203–210, 2008. doi: https://doi.org/10.1007/978-3-540-85027-4_25.

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




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

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