Time periodic solutions of one-dimensional forced Kirchhoff equations with x-dependent coefficients

In this paper, we consider the one-dimensional Kirchhoff equation with x-dependent coefficients under Dirichlet boundary conditions, which models the forced vibrations of a clamped inhomogeneous string in the presence of a time periodic external forcing with period 2π/ω and amplitude ϵ. By using the Nash-Moser iteration technique, we obtain the existence, regularity and local uniqueness of time periodic solutions with period 2π/ω and order ϵ. Such results hold for parameters (ω,ϵ) in a positive measure Cantor set that has asymptotically full measure as the amplitude ϵ goes to zero.