Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces

Abstract

We study the maximal regularity on different function spaces of the second order integro-differential equations with infinite delay $$(P)\quad\ u''(t)+\alpha {u}'(t)+\frac{d}{dt}\biggl(\int^{t}_{-\infty}b(t-s)u(s)\,ds\biggr) =Au(t)-\int^{t}_{-\infty}a