Infinitely Many Periodic Solutions of Duffing Equations with Singularities via Time Map

Abstract

We study the periodic solutions of Duffing equations with singularities $x^{\mathrm{\prime \prime }}+g(x)=p(t)$. By using Poincaré-Birkhoff twist theorem, we prove that the given equation possesses infinitely many positive periodic solutions provided that $g$ satisfies the singular condition and the time map related to autonomous system $x^{\mathrm{\prime \prime }}+g(x)=0$ tends to zero.