On locally explicit models of some nonsmooth systems

A special class of the locally explicit differential equations with nonlinear differentials that are intended to describe nonsmooth processes was introduced and studied. These equations were shown to model in a natural way behavior of some hysteresis elements. Theorems on local and global solvability of the Cauchy problem and uniqueness of solution, in particular, the generalized van Kampen theorem, were proved. New mathematical models of relay, generalized relay, stop, and closed system with relay were considered.