Periodic solutions of the euler – bernoulli quasilinear equation vibrations of a beam with an elastically fixed end

The problem of time-periodic solutions of the quasilinear Euler-Bernoulli equation of vibrations of a beam under tension along the horizontal axis is considered. The boundary conditions correspond to the cases of elastically fixed, rigidly fixed and hinged ends. The nonlinear term satisfies the nonresonance condition at infinity. Using the Schauder principle, we prove a theorem on the existence and uniqueness of a periodic solution.