Periodic solutions for completely resonant nonlinear wave equations

We consider the nonlinear string equation with Dirichlet boundary conditions $u_{xx}-u_{tt}=\varphi (u)$, with $\varphi (u)=\mathrm{\Phi }{u}^3+O(u^5)$ odd and analytic, $\mathrm{\Phi }\mathrm{\ne }0$, and we construct small amplitude periodic solutions with frequency $\text{o}$ for a large Lebesgue measure set of $\text{o}$ 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 nonresonant 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 the nonlinear wave equations $u_{xx}-u_{tt}+Mu=\varphi (u)$, $M\mathrm{\ne }0$, is that not only the P equation but also the Q equation is infinite-dimensional