On the right Lipschitz condition for differential equations with piecewise continuous right-hand sides (Q1778159)

Consider the ordinary differential equation \(dx/dt=f(t,x)\) with \(x\in \mathbb R^n\), and \(f(t,x)\) is defined in some domain and jointly continuous in \((t,x)\) everywhere with exception of a finite number of hypersurfaces. In addition, function \(f(t,x)\) satisfies a local one-sided Lipschitz condition with respect to \(x\). The existence and uniqueness of the right solutions are studied. A relation between the solutions of the differential equation and of a differential inclusion generated by the differential equation is established.