Periodic solutions of the vibrating string equation with Neumann and Dirichlet boundary conditions and a discontinuous nonlinearity

We consider a mathematical model of a vibrating string under a force that is discontinuous with respect to the state variable. It is assumed that one end of the string is fixed while the other is free. If the kernel of the operator generated by the linear part of the equation with boundary conditions and periodicity condition is zero, then the nonlinearity grows sublinearly; otherwise, it is bounded. The existence of a $2\pi$-periodic generalized solution is established by a topological method.