Newton's method and periodic solutions of nonlinear wave equations

We prove the existence of periodic solutions of the nonlinear wave equation\documentclass{article}\pagestyle{empty}\begin{document}$$ \mathop \partial \nolimits_t^2 u = \mathop \partial \nolimits_x^2 u - g(x,u), $$\end{document}satisfying either Dirichlet or periodic boundary conditions on the interval [ O , π]. The coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation. Using a version of Newton's method, we show that this equation has solutions provided the nonlinearity g ( x, u ) satisfies certain generic conditions of nonresonance and genuine nonlinearity. © 1993 John Wiley & Sons, Inc.