Analytic Solutions of Delay-Differential Equations

In 1973 Nussbaum proved that certain bounded solutions of autonomous delay-differential equations with analytic nonlinearities are themselves analytic. On the other hand, the two authors of this paper more recently showed that bounded solutions of certain delay-differential equations, again with analytic nonlinearities, can be C∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document} smooth, yet not be analytic for certain ranges of the independent variable t. In this paper we extend the 1973 results to obtain analytic solutions of a broader class of delay-differential equations, including a wide variety of nonautonomous equations. Nevertheless, there are still equations with analytic nonlinearities possessing global bounded C∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document} solutions for which analyticity is unknown. This is the case, for example, for the equation y′(t)=g(y(t-1))+εsin(t2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} y'(t)=g(y(t-1))+\varepsilon \sin (t^2) \end{aligned}$$\end{document}where g is analytic and where y=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=0$$\end{document} is a hyperbolic equilibrium when ε=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =0$$\end{document}.