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

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.