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 ) u′′(t) + αu′(t) + d dt ( t −∞ b(t− s)u(s) ds ) = Au(t)− t −∞ a(t− s)Au(s) ds+ f(t) (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), u′(0) = u′(2π), where A is a closed operator in a Banach space X, α ∈ C, and a, b ∈ L(R+). We use Fourier multipliers to characterize maximal regularity for (P ). Using known results on Fourier multipliers, we find suitable conditions on the kernels a and b under which necessary and sufficient conditions are given for the problem (P ) to have maximal regularity on L(T,X), periodic Besov spaces B p,q(T,X) and periodic Triebel–Lizorkin spaces F s p,q(T,X).