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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On invariant sets and on a theorem of Ważewski


Author: Philip Hartman
Journal: Proc. Amer. Math. Soc. 32 (1972), 511-520
MSC: Primary 34A10
DOI: https://doi.org/10.1090/S0002-9939-1972-0298091-7
MathSciNet review: 0298091
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Abstract: The first part of the paper treats the question of the existence of a solution $ x = x(t)$ of an ordinary differential equation which exists for $ t \geqq {t_0}$ and remains in a given closed set F for every assigned initial point $ ({t_0},x({t_0})) \in F$ or, in the autonomous case, $ x({t_0}) \in F$. The results involve conditions which, for the autonomous case, reduce to $ {\text{dist}}({x^0} + hf({x^0}),F)/h \to 0$ as $ h \to + 0$ for all $ {x^0} \in F$. The second part of the paper deals with theorems of the Ważewski type which, in some situations, permit the relaxation of the hypothesis that egress points are strict egress points.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0298091-7
Keywords: Invariant sets, trajectory derivative, Lyapunov, egress points
Article copyright: © Copyright 1972 American Mathematical Society