Periodic solutions of forced isochronous oscillators at resonance
Sufficient conditions are given for the existence of a \(2\pi\)-periodic solution to the equation \[ x''(t)+V^\prime(x(t))+g(x(t))=p(t),\tag{*} \] if \(V:(a,+\infty)\to{\mathbb{R}}\) is a \(2\pi/k\)-isochronous, strictly convex potential with locally Lipschitzian derivative, \(g:(a,+\infty)\to{\mathbb{R}}\) is bounded and locally Lipschitzian, and \(p\) is \(L^1_{loc}\) and \(2\pi\)-periodic. Moreover, resonance conditions of either a singular or a nonsingular kind are imposed on \(F(t,x)=V^\prime(x)+g(x)-p(t)\). A nonexistence result is given as well.
scientific article; zbMATH DE number 1836296