Infinitely Many Time‐Periodic Solutions for the Beam Equation With Concave–Convex Nonlinearities

ABSTRACT The goal of this paper is to investigate the beam equation with periodic conditions and some types of homogeneous boundary value conditions and establish the existence and multiplicity of periodic solutions for the nonlinear term involving a combination of convex and concave terms. Such a mathematical model arises from the forced vibrations of the beam. We first analyze the eigenvalues of the beam operator under certain boundary conditions, and we then set up a variational framework for such equations. Finally, by applying critical point theorems, we obtain infinitely many small‐energy and large‐energy periodic solutions.