Quasi-periodic solutions of a semilinear Liénard equation at resonance

We are concerned with the existence of quasi-periodic solutions for the following equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x'' + F_x (x,t)x' + \omega ^2 x + \phi (x,t) = 0,$$ \end{document} where F and φ are smooth functions and 2π-periodic in t, ω > 0 is a constant. Under some assumptions on the parities of F and φ, we show that the Dancer’s function, which is used to study the existence of periodic solutions, also plays a role for the existence of quasi-periodic solutions and the Lagrangian stability (i.e. all solutions are bounded).