$\theta$-metric function

I. G. Tsar'kov

Izvestiya. Mathematics , Tome 88 (2024) no. 2, pp. 369-388.

Résumé

We study approximative properties of sets as a function of the rate of variation of the distance function defined in terms of some continuous functional (in lieu of a metric). As an application, we prove non-uniqueness of approximation by non-convex subsets of Hilbert spaces with respect to special continuous functionals. Results of this kind are capable of proving non-uniqueness solvability for gradient-type equations.

Détail

Keywords: asymmetric space, $\theta$-metric function, minimization of functionals, differential equation, $\theta$-metric projection.
Mots-clés : non-unique solvability

Bibliographie

[1] Ş. Cobzaş, Functional analysis in asymmetric normed spaces, Front. Math., Birkhäuser/Springer Basel AG, Basel, 2013 | DOI | MR | Zbl

[2] S. Cobzaş, “Separation of convex sets and best approximation in spaces with asymmetric norm”, Quaest. Math., 27:3 (2004), 275–296 | DOI | MR | Zbl

[3] W. B. Moors, “Nearly Chebyshev sets are almost convex”, Set-Valued Var. Anal., 26:1 (2018), 67–76 | DOI | MR | Zbl

[4] B. Ricceri, “Multiplicity theorems involving functions with non-convex range”, Studia Univ. Babes-Bolyai, Math., 68:1 (2023), 125–137 | DOI

[5] I. G. Tsar'kov, “The distance function and boundedness of diameters of the nearest elements”, Modern methods in operator theory and harmonic analysis, Springer Proc. Math. Stat., 291, Springer, Cham, 2019, 263–272 | DOI | MR

[6] A. R. Alimov and I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russian Math. Surveys, 71:1 (2016), 1–77 | DOI

[7] V. S. Balaganskii and L. P. Vlasov, “The problem of convexity of Chebyshev sets”, Russian Math. Surveys, 51:6 (1996), 1127–1190 | DOI

[8] A. R. Alimov and I. G. Tsar'kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, J. Math. Sci. (N.Y.), 217:6 (2016), 683–730 | DOI

[9] A. R. Alimov and I. G. Tsar'kov, “Ball-complete sets and solar properties of sets in asymmetric spaces”, Results Math., 77:2 (2022), 86 | DOI | MR | Zbl

[10] A. R. Alimov and I. G. Tsar'kov, Geometric approximation theory, Springer Monogr. Math., Springer, Cham, 2021 | DOI | MR | Zbl

[11] A. R. Alimov and I. G. Tsar'kov, “Suns, moons, and ${\mathring B}$-complete sets in asymmetric spaces”, Set-Valued Var. Anal., 30:3 (2022), 1233–1245 | DOI | MR | Zbl

[12] A. R. Alimov and I. G. Tsarkov, “$\mathring B$-complete sets: approximative and structural properties”, Siberian Math. J., 63:3 (2022), 412–420 | DOI

[13] I. G. Tsar'kov, “Singular sets of surfaces”, Russ. J. Math. Phys., 24:2 (2017), 263–271 | DOI | MR | Zbl

[14] I. G. Tsar'kov, “Geometry of the singular set of hypersurfaces and the eikonal equation”, Russ. J. Math. Phys., 29:2 (2022), 240–248 | DOI | MR | Zbl

[15] I. G. Tsar'kov, “Approximative and structural properties of sets in asymmetric spaces”, Izv. Math., 86:6 (2022), 1240–1253 | DOI

[16] A. R. Alimov and I. G. Tsar'kov, “Some classical problems of geometric approximation theory in asymmetric spaces”, Math. Notes, 112:1 (2022), 3–16 | DOI | MR

[17] I. G. Tsar'kov, “Smoothness of solutions of the eikonal equation and regular points of their level surfaces”, Russ. J. Math. Phys., 30:2 (2023), 259–269 | DOI | MR | Zbl

[18] V. Donjuán and N. Jonard-Pérez, “Separation axioms and covering dimension of asymmetric normed spaces”, Quaest. Math., 43 (4), 467–491 | DOI | MR | Zbl

[19] I. G. Tsar'kov, “Approximative compactness and nonuniqueness in variational problems, and applications to differential equations”, Sb. Math., 202:6 (2011), 909–934 | DOI

[20] I. G. Tsar'kov, “Nonunique solvability of certain differential equations and their connection with geometric approximation theory”, Math. Notes, 75:2 (2004), 259–271 | DOI