Miailou NAPO, Mohamado KIEMA, Issa ZABSONRE

LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY

Anahtar Kelimeler:

Cosine family, Mild and strict solutions, second order differential equations, $\alpha$-norm, delay differential equation.

LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY

In this work, we study the existence and regularity of solutions for some second order differential equations with infinite 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, \emph{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. Secondly, we give sufficient conditions ensuring the existence of strict solutions.

Keywords:

Cosine family, second order differential equations, $\alpha$-norm, delay differential equation.,

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