Periodic Solutions for the Semilinear Euler–Bernoulli Beam Equation With Hinged and Elastically Fixed Ends

We consider the forced vibration of Euler–Bernoulli beam equation with hinged and elastically fixed ends, and study the existence of multiple periodic solutions for such an equation with the nonlinear term having superliner growth but not requiring its homogeneity and monotonicity. At the first step, we need to analyze the asymptotic formula of the eigenvalues and eigenfunctions for the corresponding Sturm–Liouville problem. Then we investigate the fundamental properties of the linear beam operator and the corresponding functional. Finally, in view of the effect of inhomogeneous term, we shall construct a modified functional and make use of the minimax method and Morse index to obtain the existence of infinitely many periodic solutions.