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.