Families of periodic solutions of resonant PDEs

We construct some families of small amplitude periodic solutions close to a completely resonant equilibrium point of a semilinear reversible partial d ifferential equation. To this end, we construct, using averaging methods, a suitable map from the configuration space to itself. We prove that to each nondegenerate zero of such a map there corresponds a family of small ampli tude periodic solutions of the system. The proof is based on Lyapunov-Schmi dt decomposition. This establishes a relation between Lyapunov-Schmidt deco mposition and averaging theory that could be interesting in itself. As an a pplication, we construct countable many families of periodic solutions of t he nonlinear string equation u(tt) - u(xx) +/- u(3) = 0 (and of its perturb ations) with Dirichlet boundary conditions. We also prove that the fundamen tal periods of solutions belonging to the n(th) family converge to 2 pi /n when the amplitude tends to zero.