Mild solutions to integro-differential equations in Banach spaces

Well‐posedness of second order differential equations with memory

“…On the other hand, similar methods have been used by several authors to give necessary conditions for the existence and uniqueness of periodic mild solutions of second order differential equations in Banach spaces, see for instance [4,6,18,20,21,28]. In the case of mild solution of second order differential equations on the real line, we refer to [28,30] and the references therein, and for systems of integro-differential equations on the line, we mention here for instance to [15]. However, to the best of our knowledge, this problem has not been considered in the case of integro-differential equations in the form of (1.3).…”