Maximal regularity of second order delay equations in Banach spaces

Using known Cα-multiplier result, we give necessary and sufficient conditions for the second order delay equations: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u''(t) = Au(t) + Fu_t + Gu'_t + f(t), t \in \mathbb{R} $$\end{document} to have maximal regularity in Hölder continuous function spaces Cα(ℝ, X), where X is a Banach space, A is a closed operator in X, F, G ∊ ℒ(C([−r, 0],X), X) are delay operators for some fixed r > 0.