In this paper we examine periodic integrodifferential equations in Banach spaces. When the cone is regular, we prove two existence theorems for the extremal solutions in the order interval determined by an upper and a lower solution. Both theorems use only the order structure of the problem and no compactness condition is assumed. In the last section we ask the cone to be only normal but we impose a compactness condition using the ball measure of noncompactness. We obtain the extremal solutions...

We study the maximal regularity on different function spaces of the second order integro-differential equations with infinite delay $\left(P\right)u^{\text{'}\text{'}}\left(t\right)+\alpha u^{\text{'}}\left(t\right)+d/dt\left({\int }_{-\infty }^{t}b(t-s)u\left(s\right)ds\right)=Au\left(t\right)-{\int }_{-\infty }^{t}a(t-s)Au\left(s\right)ds+f\left(t\right)$ (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 ∈ L¹(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...