Local existence and regularity of solutions for some second order differential equation with infinite delay

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Résumé

In this work, we study the existence and regularity of solutions for some second order dierential equations with innite delay in Banach spaces. We suppose that the undelayed part admits a cosine operator in the sense given
by Da Prato and Giusi, [ G. Da Prato and E. Giusi, Una caratterizzazione dei generatori di funzioni coseno astratte, Bollettino dell'Unione Matematica Italiana, 22, 357-362, (1967)]. The delayed part is assumed to be locally Lipschitz.
Firstly, we show the existence of the mild solutions. Results are obtained by using Schauder and Banach-Piccard xed point theorems. We also prove that the mild solution continuously depends on initial data. Secondly, we give suf-
cient conditions ensuring the existence of strict solutions. Last section is devoted to an application.

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