Existence and uniqueness of Carathéodory and Filippov solutions for discontinuous systems of differential equations

Abstract

We use essential limits inferior and superior of the nonlinear part of a discontinuous ODE to introduce some novel transversality conditions which imply that Filippov solutions are Carathéodory solutions. We also prove some uniqueness criteria based on different Lipschitz conditions on different parts of the domain separated from one another by boundaries which satisfy certain transversality conditions.