Periodic solutions for a beam equation with concave-convex nonlinearities

We are concerned with the time-periodic solutions of the beam equation with concave-convex nonlinearities which describes the forced vibrations of the beam. In general, the super-linear term plays a dominated role at $ \infty $ and the sub-linear term plays a dominated role at $ 0 $. By setting a variational framework, we take advantage of the mini-max principle to obtain two sequences of time-periodic solutions with large energy and small energy, respectively.