Invariant tori and boundedness of solutions of non-smooth oscillators with Lebesgue-integrable forcing term

Since Littlewood works in the 1960s, the boundedness of solutions of Duffing-type equations x center dot+g(x)=p(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{x}+g(x)=p(t)$$\end{document} has been extensively investigated. More recently, some researches have focused on the family of non-smooth forced oscillators x center dot+sign(x)=p(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \ddot{x}+\textrm{sign}(x)=p(t)$$\end{document}, mainly because it represents a simple limit scenario of Duffing-type equations for when g is bounded. Here, we provide a simple proof for the boundedness of solutions of the non-smooth forced oscillator in the case that the forcing term p(t) is a T-periodic Lebesgue-integrable function with vanishing average. We reach this result by constructing a sequence of invariant tori whose union of their interiors covers all the (t,x,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t,x,\dot{x})$$\end{document}-space, (t,x,x)is an element of S1xR2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t,x,\dot{x})\in {\mathbb {S}}<^>1\times {\mathbb {R}}<^>2$$\end{document}.