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

Authors : Miailou NAPO, Mohamado KIEMA, Issa ZABSONRE

Pages : 161-182

Doi:10.33773/jum.1233884

View : 47 | Download : 52

Publication Date : 2023-07-31

Article Type : Research Paper

Abstract :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, insert ignore into journalissuearticles values(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, Mild and strict solutions, second order differential equations, \\alpha norm, delay differential equation