Unique solutions for a class of discontinuous differential equations

Abstract:

This paper is concerned with the Cauchy Problem \[ \dot x\left ( t \right ) = f\left ( {t,x\left ( t \right )} \right ),\quad x\left ( {{t_0}} \right ) = {x_0} \in {\mathbb {R}^n},\] where the vector field $f$ may be discontinuous with respect to both variables $t,x$. If the total variation of $f$ along certain directions is locally finite, we prove the existence of a unique solution, depending continuously on the initial data.