Existence of solutions for a nonlinear wave equation

We prove the existence of a strong solution of a periodic-Dirichlet problem for the semilinear wave equation with irrational period and with nonlinearity satisfying some general growth conditions locally around $0$. We construct a new variational method, called a dual method, and using relations between critical points and critical values of the primal action and the dual action functionals we prove that the solution exists. The dual functional which we define is different from the ones known so far in that it depends on two dual variables.