×

The structure of the set of local minima of functions in various spaces. (English. Russian original) Zbl 1429.54023

Sib. Math. J. 60, No. 3, 398-404 (2019); translation from Sib. Mat. Zh. 60, No. 3, 518-526 (2019).
The authors study properties of sets of strict local extrema of real-valued functions defined on generalized metric spaces, such as being meager or a countable set. Most of these results were previously known for functions defined on metric spaces. Here, functions defined on \(f\)-quasimetric spaces are considered. For the definition of \(f\)-quasimetric spaces see [A. V. Arutyunov et al., Topology Appl. 221, 178–194 (2017; Zbl 1377.54028)].

MSC:

54C30 Real-valued functions in general topology
54E99 Topological spaces with richer structures

Citations:

Zbl 1377.54028
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arutyunov A. V. and Greshnov A. V., “(<Emphasis Type=”Italic“>q1, <Emphasis Type=”Italic“>q2)-Quasimetric spaces. Covering mappings and coincidence points,” Izv. Math., vol. 82, no. 2, 245-272 (2018). · Zbl 1401.54023 · doi:10.1070/IM8546
[2] Arutyunov A. V. and Zhukovskiy S. E., “Variational principles in nonlinear analysis and their generalization,” Math. Notes, vol. 103, no. 6, 1014-1019 (2018). · Zbl 1432.49019 · doi:10.1134/S0001434618050383
[3] Behrends E., Geschke S., and Natkaniec T., “Functions for which all points are local extrema,” Real Anal. Exchange, vol. 33, no. 2, 467-470 (2007). · Zbl 1170.26002 · doi:10.14321/realanalexch.33.2.0467
[4] Balcerzak M., Popĺawski M., and Wydka J., “Local extrema and nonopenness points of continuous functions,” Amer. Math. Monthly, vol. 124, no. 5, 436-443 (2017). · Zbl 1391.26012 · doi:10.4169/amer.math.monthly.124.5.436
[5] Arutyunov A. V., “Second-order conditions in extremal problems. The abnormal points,” Trans. Amer. Math. Soc., vol. 350, no. 11, 4341-4365 (1998). · Zbl 0922.49017 · doi:10.1090/S0002-9947-98-01775-9
[6] Arutyunov A. V. and Vinter R. B., “A simple ‘finite approximations’ proof of the Pontryagin maximum principle under reduced differentiability hypotheses,” Set-Valued Anal., vol. 12, 5-24 (2004). · Zbl 1046.49014 · doi:10.1023/B:SVAN.0000023406.16145.a8
[7] Arutyunov A. V., Greshnov A. V., Lokutsievskii L. V., and Storozhuk K. V., “Topological and geometrical properties of spaces with symmetric and nonsymmetric <Emphasis Type=”Italic“>f-quasimetrics,” Topology Appl., vol. 221, 178-194 (2017). · Zbl 1377.54028 · doi:10.1016/j.topol.2017.02.035
[8] Stoltenberg R. A., “On quasi-metric spaces,” Duke Math. J., vol. 36, no. 1, 65-71 (1969). · Zbl 0176.51902 · doi:10.1215/S0012-7094-69-03610-2
[9] Birkhoff G., “A note on topological groups,” Compositio Math., vol. 3, 427-430 (1936). · Zbl 0015.00702
[10] Frink A. H., “Distance functions and the metrization problem,” Bull. Amer. Math. Soc., vol. 43, 133-142 (1937). · JFM 63.0571.03 · doi:10.1090/S0002-9904-1937-06509-8
[11] Storozhuk K. V., “Strong extrema of functions on quasi-metric and compact spaces,” Topology Appl., vol. 250, 37-47 (2018). · Zbl 1408.54003 · doi:10.1016/j.topol.2018.10.001
[12] Kunzi H. P. A., “On strongly quasi-metrizable spaces,” Arch. Math., vol. 41, 57-63 (1983). · Zbl 0504.54028 · doi:10.1007/BF01193823
[13] Ribeiro H., “Sur les espaces ametrique faible,” Portugal. Math., vol. 4, no. 1, 21-40 (1943). · Zbl 0028.19103
[14] Alexandroff P. S. and Urysohn P. S., Mémoire sur les Espaces Topologiques Compacts, Nauka, Moscow (1971). · JFM 49.0405.02
[15] Engelking R., General Topology, Heldermann Verlag, Berlin (1989). · Zbl 0684.54001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.