Bifurcation and multiplicity results for periodic solutions of a damped wave equation in a thin domain

We study the bifurcation problem for periodic solutions of a nonautonomous damped wave equation defined in a thin domain. Here the bifurcation parameter is represented by the thinness ε>0 of the considered domain. This study has as starting point the existence result of periodic solutions already stated by the authors for this equation and it makes use of the condensivity properties of the associated Poincaré map and its linearization around these solutions. We establish sufficient conditions to guarantee that ε=0 is or not a bifurcation point and a related multiplicity result. These results are in the spirit of those given by Krasnoselʹskii and they are obtained by using the topological degree theory for k-condensing operators.