A small twist theorem and boundedness of solutions for semilinear Duffing equations at resonance

We obtain a new variant of Moser’s small twist theorem and apply this new version to investigate the boundedness of solutions for the following semilinear Duffing equation ẍ+n2x+g(x)=p(t), where pp is a 2π2π-periodic smooth function and lim|x|→∞x−1g(x)=0lim|x|→∞x−1g(x)=0. We obtain some sharp sufficient conditions for the boundedness of all   solutions to the above equation at resonance. Unlike many existing results in the literature where the function gg is required to be a bounded function with asymptotic limits, our main results here allow gg be unbounded or oscillatory without asymptotic limits.