On sets of discontinuous solutions of nonlinear differential equations (Q1897957)

The author studies the properties of the pencils of discontinuous solutions for the system of nonlinear differential equations \(\dot x = f(t,x,v,V) + B(t,x,v,V) \dot v(t)\), where \(f(t,x,v,V)\) is continuous with respect to the totality of its variables and Lipschitz-continuous with respect to \(x\), \(B(t,x,v,V)\) is an \(n \times m\)-dimensional matrix- function whose entries are continuous with respect to the totality of its variables and Lipschitz-continuous with respect to \(x\), \(t \in [t_0, \nu]\), \(x \in \mathbb{R}^n\), \(v \in \mathbb{R}^m\), \(v(t) \in BV_m [t_0, \nu]\) (here \(BV_m [t_0, \nu]\) stands for the Banach space of \(l\)- dimensional functions of bounded variation). The problem arising during the study of equation (1) is connected with multiplication of distributions in the second term of the right-hand side of the equation (1) and it is resolved with the use of the closing of the set of smooth solutions generated by the absolutely continuous vector-functions \(v(t)\).