Periodic solutions of nonlinear vibrating beams

Abstract

The aim of this paper is to prove new existence and multiplicity results for periodic semilinear beam equation with a nonlinear time-independent perturbation in case the period is not prescribed. Since the spectrum of the linear part varies with the period, the solvability of the equation depends crucially on the period which can be chosen as a free parameter. Since the period of the external forcing is generally unknown a priori, we consider the following natural problem. For a given time-independent nonlinearity, find periods $T$ for which the equation is solvable for any $T$-periodic forcing. We will also deal with the existence of multiple solutions when the nonlinearity interacts with the spectrum of the linear part. We show that under certain conditions multiple solutions do exist for any small forcing term with suitable period $T$. The results are obtained via generalized Leray-Schauder degree and reductions to invariant subspaces.