Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions

We consider the nonlinear string equation with Dirichlet boundary conditions u(tt) - u(xx) = phi(u), with phi(u) = Phi u(3) + O(u(5)) odd and analytic, Phi not equal 0, and we construct small amplitude periodic solutions with frequency omega for a large Lebesgue measure set of omega close to 1. This extends previous results where only a zero-measure set of frequencies could be treated ( the ones for which no small divisors appear). The proof is based on combining the Lyapunov-Schmidt decomposition, which leads to two separate sets of equations dealing with the resonant and non-resonant Fourier components, respectively the Q and the P equations, with resummation techniques of divergent powers series, allowing us to control the small divisors problem. The main difficulty with respect to the nonlinear wave equations u(tt) - u(xx) + Mu = phi(u), M not equal 0, is that not only the P equation but also the Q equation is infinite-dimensional.