Periodic solutions of the telegraph equation with a discontinuous nonlinearity

We consider telegraph equations with a variable inner energy, discontinuous by phase, and the homogeneous Dirichlet boundary condition. Question of existence of general periodic solutions in the resonant case, when the operator created by a linear part of the equation with the homogeneous Dirichlet boundary condition and the condition of periodicity has a non zero kernel, and nonlinearity appearing in the equation is limited. We obtained an existence theorem for the general periodic solution bt means of the topological method. The proof is based on the Leray–Schauder principle for convex compact mappings. The main difference from similar results of other authors is an assumption that there are breaks in the phase variable of the inner energy of the telegraph equation.