Some multiplicity results for periodic solutions of a Rayleigh differential equation (Q2756182)

The authors study the existence and multiplicity of \(T\)-periodic solutions to a Rayleigh equation of the type \(x'' + f(x') + g(t,x,x') = \overline{p},\) where \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function, \(g: [0,T]\times \mathbb{R}^{2} \rightarrow \mathbb{R}\) is a Carathéodory function and \(\overline{p} \in \mathbb{R}\). The hypotheses on the nonlinearities include some classical situations such as Ambrosetti-Prodi-type problems or pendulum-type equations. In particular, they establish conditions under which the set of \(\overline{p}\) for which the previous equation has at least one or at least two periodic solutions is an interval (bounded or unbounded). The main tools used in the proofs are upper and lower solutions and topological degree.