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 Trudy Inst. Mat. i Mekh. UrO RAN, 2007, Volume 13, Number 2, Pages 167–183 (Mi timm99)

Euler's broken lines in systems with Carathéodory conditions

D. V. Khlopin

Abstract: We consider Euler's broken lines in a system with its right-hand side measurable in time and investigate their convergence to trajectories of the system. Counterexamples are given that show that partitions with a small diameter do not guarantee the proximity to the funnel of trajectories. For any Carathéodory function, it is suggested to equip the set of closed subsets of the time interval with a metric. We prove that, under conditions close to Carathéodory ones, the convergence with respect to the metric guarantees the convergence of Euler's broken lines to the funnel of solutions of the system. As a consequence, it is shown that if the right-hand side is continuous and the sublinear growth condition is satisfied, then a sufficiently small diameter of the partition guarantees the proximity of Euler's broken line to the funnel of solutions of the system.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2007, 259, suppl. 2, S141–S158

UDC: 517.928.1+517.929.8

Citation: D. V. Khlopin, “Euler's broken lines in systems with Carathéodory conditions”, Trudy Inst. Mat. i Mekh. UrO RAN, 13, no. 2, 2007, 167–183; Proc. Steklov Inst. Math. (Suppl.), 259, suppl. 2 (2007), S141–S158

Citation in format AMSBIB
\Bibitem{Khl07} \by D.~V.~Khlopin \paper Euler's broken lines in systems with Carath\'eodory conditions \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2007 \vol 13 \issue 2 \pages 167--183 \mathnet{http://mi.mathnet.ru/timm99} \elib{https://elibrary.ru/item.asp?id=12040779} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2007 \vol 259 \issue , suppl. 2 \pages S141--S158 \crossref{https://doi.org/10.1134/S0081543807060090} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-38949133162} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. V. Khlopin, “Lomanye Eilera i vremennye shkaly v usloviyakh Karateodori”, Tr. IMM UrO RAN, 14, no. 4, 2008, 159–171
2. D. V. Khlopin, “Skhodimost lomanykh Eilera v usloviyakh Karateodori”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2008, no. 2, 163–164
3. D. V. Khlopin, “Lomanye Eilera i diametr razbieniya”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 7:4 (2014), 102–112
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