Application of p-regularity theory to the Duffing equation

Abstract

The paper studies a solution existence problem of the nonlinear Duffing equation of the form F(x,μ,β)=x″+x+μx3−βsint=0,β>0,μ≠0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ F(x,\\mu, \\beta)=x''+x+\\mu x^{3}-\\beta\\sin t = 0,\\quad \\beta > 0, \\mu\\neq0, $$\\end{document} where F: mathcal{C}^{2}[0,2pi]timesmathbb{R}times mathbb{R}rightarrowmathcal{C}[0,2pi] and x(0)=x(2pi)=0 using the p-regularity theory.