{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T23:57:52Z","timestamp":1740182272137,"version":"3.37.3"},"reference-count":0,"publisher":"Tambov State University - G.R. Derzhavin","issue":"128","funder":[{"DOI":"10.13039\/501100002261","name":"\u0420\u043e\u0441\u0441\u0438\u0439\u0441\u043a\u0438\u0439 \u0424\u043e\u043d\u0434 \u0424\u0443\u043d\u0434\u0430\u043c\u0435\u043d\u0442\u0430\u043b\u044c\u043d\u044b\u0445 \u0418\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u0439","doi-asserted-by":"publisher","award":["19-01-00080_\u0430","20-31-70013_\u0421\u0442\u0430\u0431\u0438\u043b\u044c\u043d\u043e\u0441\u0442\u044c"],"award-info":[{"award-number":["19-01-00080_\u0430","20-31-70013_\u0421\u0442\u0430\u0431\u0438\u043b\u044c\u043d\u043e\u0441\u0442\u044c"]}],"id":[{"id":"10.13039\/501100002261","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001663","name":"Volkswagen Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019]]},"abstract":"<jats:p>We study the question of the existence of a solution to the Cauchy problem for a di\ufb00erential equation unsolved with respect to the derivative of the unknown function. Di\ufb00erential equations generated by twice continuously di\ufb00erentiable mappings are considered. We give an example showing that the assumption of regularity of the mapping at each point of the domain is not enough for the solvability of the Cauchy problem. The concept of uniform regularity for the considered mappings is introduced. It is shown that the assumption of uniform regularity is su\ufb03cient for the local solvability of the Cauchy problem for any initial point in the class of continuously di\ufb00erentiable functions. It is shown that if the mapping de\ufb01ning the di\ufb00erential equation is majorized by mappings of a special form, then the solution of the Cauchy problem under consideration can be extended to a given time interval. The case of the Lipschitz dependence of the mapping de\ufb01ning the equation on the phase variable is considered. For this case, estimates of non-extendable solutions of the Cauchy problem are found. The results are compared with known ones. It is shown that under the assumptions of the proved existence theorem, the uniqueness of a solution may fail to hold. We provide examples llustrating the importance of the assumption of uniform regularity.<\/jats:p>","DOI":"10.20310\/2686-9667-2019-24-128-376-383","type":"journal-article","created":{"date-parts":[[2020,1,24]],"date-time":"2020-01-24T13:14:10Z","timestamp":1579871650000},"page":"376-383","source":"Crossref","is-referenced-by-count":0,"title":["On the existence of a continuously di\ufb00erentiable solution to the Cauchy problem for implicit di\ufb00erential equations"],"prefix":"10.20310","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4595-6685","authenticated-orcid":false,"given":"Zukhra T.","family":"Zhukovskaya","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":[[2019]]},"container-title":["Russian Universities Reports. Mathematics"],"original-title":["\u041e \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u043e\u0432\u0430\u043d\u0438\u0438 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u043e\u0433\u043e \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u0447\u0438 \u041a\u043e\u0448\u0438 \u0434\u043b\u044f \u043d\u0435\u044f\u0432\u043d\u044b\u0445 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u044b\u0445 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0439"],"language":"ru","deposited":{"date-parts":[[2020,7,14]],"date-time":"2020-07-14T22:06:31Z","timestamp":1594764391000},"score":1,"resource":{"primary":{"URL":"https:\/\/elibrary.ru\/item.asp?id=41720279"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019]]},"references-count":0,"journal-issue":{"issue":"128","published-online":{"date-parts":[[2019]]},"published-print":{"date-parts":[[2019]]}},"URL":"https:\/\/doi.org\/10.20310\/2686-9667-2019-24-128-376-383","relation":{},"ISSN":["2686-9667"],"issn-type":[{"type":"print","value":"2686-9667"}],"subject":[],"published":{"date-parts":[[2019]]}}}