Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics

Iacopo P. Longo; Sylvia Novo; Rafael Obaya

doi:10.3934/dcds.2019224

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2019, Volume 39, Issue 9: 5491-5520. Doi: 10.3934/dcds.2019224

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Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics

Departamento de Matemática Aplicada, E.I. Industriales, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid, Spain

^* Corresponding author: Rafael Obaya
^* Corresponding author: Rafael Obaya

Received: December 31, 2018

Published: May 2019

Partly supported by MINECO/FEDER under project MTM2015-66330-P and EU Marie-Skłodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-ITN-2014 643073 CRITICS)

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Abstract

We study some already introduced and some new strong and weak topologies of integral type to provide continuous dependence on continuous initial data for the solutions of non-autonomous Carathéodory delay differential equations. As a consequence, we obtain new families of continuous skew-product semiflows generated by delay differential equations whose vector fields belong to such metric topological vector spaces of Lipschitz Carathéodory functions. Sufficient conditions for the equivalence of all or some of the considered strong or weak topologies are also given. Finally, we also provide results of continuous dependence of the solutions as well as of continuity of the skew-product semiflows generated by Carathéodory delay differential equations when the considered phase space is a Sobolev space.

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Mathematics Subject Classification: Primary: 34A34, 37B55, 34A12, 34F05.

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