Time Periodic Solutions of One-Dimensional Forced Kirchhoff Equation with Sturm–Liouville Boundary Conditions

Abstract

This paper is concerned with the time periodic solutions for the Sturm–Liouville boundary value problem of a one-dimensional Kirchhoff equation in presence of a time periodic external forcing with period $$2\pi /\omega $$ and amplitude $$\epsilon $$. Such a model arises from the forced vibrations of a bounded string in which the dependence of the tension on the deformation cannot be neglected. By using the Nash–Moser iteration technique, we obtain the existence, regularity and local uniqueness of time periodic solutions with period $$2\pi /\omega $$ and order $$\epsilon $$. Such results hold for parameters $$(\omega ,\epsilon )$$ in a positive measure Cantor set that has asymptotically full measure as $$\epsilon $$ goes to zero.