Periodic Solutions for a Class of Semilinear Euler-Bernoulli Beam Equations with Variable Coefficients

This paper is devoted to the study of periodic solutions for a class of semilinear Euler-Bernoulli beam equations with variable coefficients. Such a mathematical model is used to describe the infinitesimal, free, undamped in-plane bending vibrations of a thin straight elastic beam. When the frequency is rational, we acquire some fundamental properties of the variable coefficients beam operator and in particular prove that its inverse operator is compact on its range. Based on these properties, we obtain the existence of periodic solutions when the nonlinear term is monotone and bounded.