{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T23:58:07Z","timestamp":1740182287594,"version":"3.37.3"},"reference-count":6,"publisher":"Tambov State University - G.R. Derzhavin","issue":"144","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023]]},"abstract":"<jats:p>This work is devoted to the study of the covering property of linear and nonlinear mappings of Banach spaces. We consider linear continuous operators acting from one Banach space to another. For a given operator, it is shown that for any point y_0 from the relative interior of the image of a given convex closed cone there exists a conical neighborhood of y_0, with respect to which the given operator has the covering property at zero with a covering constant depending on the point y0: We provide an example showing that for a linear continuous operator the covering property with respect to the image of a given cone at zero may fail, i. e. the statement of Banach\u2019s theorem on an open mapping may not hold for restrictions of linear continuous operators to closed convex cones. We obtain a corollary of the obtained theorem for the case when the target space is finite-dimensional. Moreover, nonlinear twice differentiable mappings of Banach spaces are considered. For them, conditions for local covering along a certain curve with respect to a given cone are presented. The corresponding sufficient conditions are formulated in terms of 2 -regular directions. They remain meaningful even in the case of degeneracy of the first derivative of the mapping under consideration at a given point.<\/jats:p>","DOI":"10.20310\/2686-9667-2023-28-144-361-370","type":"journal-article","created":{"date-parts":[[2023,12,15]],"date-time":"2023-12-15T11:04:32Z","timestamp":1702638272000},"page":"361-370","source":"Crossref","is-referenced-by-count":0,"title":["Generalization of Banach\u2019s theorem for cones and covering along curves"],"prefix":"10.20310","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7326-7492","authenticated-orcid":false,"given":"Aram V.","family":"Arutyunov","sequence":"first","affiliation":[{"name":"V.A. Trapeznikov Institute of Control Sciences of RAS"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2686-4654","authenticated-orcid":false,"given":"Sergey E.","family":"Zhukovskiy","sequence":"additional","affiliation":[{"name":"V.A. Trapeznikov Institute of Control Sciences of RAS"}]}],"member":"8278","published-online":{"date-parts":[[2023]]},"reference":[{"key":"ref0","unstructured":"[1] A.N. Kolmogorov, S.V. Fomin, Elements of the Theory of Functions and Functional Analysis, I, II, Dover Publications, Mineola, New York, 1957, 1961."},{"key":"ref1","unstructured":"[2] I. Ekeland, R. Temam, Convex Analysis and Variational Problems, SIAM, Philadelphia, 1976, 402 pp."},{"key":"ref2","doi-asserted-by":"publisher","unstructured":"[3] A.V. Dmitruk, A.A. Milyutin, N.P. Osmolovskii, \u201cLyusternik\u2019s theorem and the theory of extrema\u201d, Russian Math. Surveys, 35:6 (1980), 11\u201351.","DOI":"10.1070\/rm1980v035n06abeh001973"},{"key":"ref3","unstructured":"[4] L. Lusternik, \u201cSur les extr\u0491em\u0491es relatifs des fonctionnelles\u201d, Mat. Sb., 41:3 (1934), 390\u2013401 (In Russian)."},{"key":"ref4","unstructured":"[5] A.D. Ioffe, V.M. Tikhomirov, Theory of Extremal Problems, Science Publ., Moscow, 1974 (In Russian)."},{"key":"ref5","doi-asserted-by":"publisher","unstructured":"[6] E.R. Avakov, \u201cTheorems on estimates in the neighborhood of a singular point of a mapping\u201d, Math. Notes, 47:5 (1990), 425\u2013432.","DOI":"10.1007\/bf01158081"}],"container-title":["Russian Universities Reports. Mathematics"],"original-title":["\u041a\u043e\u043d\u0443\u0441\u043d\u043e\u0435 \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u0435 \u0442\u0435\u043e\u0440\u0435\u043c\u044b \u0411\u0430\u043d\u0430\u0445\u0430 \u0438 \u043d\u0430\u043a\u0440\u044b\u0432\u0430\u043d\u0438\u0435 \u0432\u0434\u043e\u043b\u044c \u043a\u0440\u0438\u0432\u044b\u0445"],"language":"ru","deposited":{"date-parts":[[2023,12,15]],"date-time":"2023-12-15T11:04:40Z","timestamp":1702638280000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.elibrary.ru\/item.asp?id=55367371"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023]]},"references-count":6,"journal-issue":{"issue":"144","published-online":{"date-parts":[[2023]]},"published-print":{"date-parts":[[2023]]}},"URL":"https:\/\/doi.org\/10.20310\/2686-9667-2023-28-144-361-370","relation":{},"ISSN":["2686-9667"],"issn-type":[{"type":"print","value":"2686-9667"}],"subject":[],"published":{"date-parts":[[2023]]}}}