Mathematics

We prove existence of small amplitude $2\pi \slash \om$-periodic solutions of the completely resonant nonlinear wave equation u_{tt} - u_{xx} + f (x, u ) = 0, u ( t, 0 )= u( t, \pi ) = 0 where f(x,0)= f_u (x,0) =0, for any frequency $ \om $ belonging to a Cantor like set of positive measure. The proof is based on a Lyapunov-Schmidt reduction and on a variant of Nash-Moser Implicit Function Theorems to overcome the inherent "small divisor problem".