Paths of nonlinear systems with a relay type nonlinearity

A system of linear differential equations with one discontinuous nonlinearity of the relay type has a unique solution for almost all initial condition. For some initial conditions, however, a family of solutions may exist which are not piecewisesmooth time functions. The conventional methods to solve differential equations prevent asserting their existence as well as finding such solutions. A method to obtain such solutions is suggested; an example is given.