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We shall study the existence of time-periodic solutions for a semilinear wave equation with a given time-independent nonlinear perturbation and small forcing. Since the distribution of eigenvalues of the linear part varies with the period, the solvability of the problem depends essentially on the frequency. The main idea of this paper is to consider the situation where the period is not prescribed and hence treated as a parameter. The description of the distribution of eigenvalues as a function of the period enables us to show that under certain conditions the interaction between the nonlinearity and the spectrum of the wave operator induces multiple solutions. Our basic new result states that the autonomous equation admits at least two nontrivial solutions (free vibrations) for a restricted (but infinite) set of periods such that the nonlinearity interacts with one simple eigenvalue. As a corollary we prove that the semilinear wave equation with time-independent nonlinearity and small forcing admits an infinite sequence of pairs of periodic solutions with corresponding period tending to zero. The results are obtained via generalized topological degree theory.

A new geometric proof of the spectral theorem for unbounded selfadjoint operators A in a Hilbert space H is given based on a splitting of A in positive and negative parts A + ≥ 0 and A − ≤ 0. For both operators A + and A − the spectral family can be defined immediately and then put together to become the spectral family of A. Of course crucial methods and results of [Lei79] are used.