Periodic Solutions of a Quasilinear Wave Equation

We study the properties of wave operators satisfying the periodicity condition with respect to time and homogeneous boundary conditions of the third kind and of Dirichlet type. We prove the existence of a nontrivial periodic (in time) $\sin$-Gordon solution with homogeneous boundary conditions of the third kind and of Dirichlet type. We obtain theorems on the existence of periodic solutions of a quasilinear wave equation with variable (in $x$) coefficients and a boundary condition of the third kind.