Converging multistep methods for initial value problems involving multivalued maps

Every consistent and strongly stable multistep method of stepnumberk yields a solution, of the setvalued initial value problem \(\dot y \in F(t,y),y(t_0 ) = y_0 \) . The setF(t, z) is assumed to be nonvoid, convex and closed. Upper semicontinuity of F with respect to both variables is not required everywhere. If the initial value problem is uniquely solvable, the solutions of the multistep method will converge to the solution of the continuous problem. These results carry over to functional differential equations \(\dot y \in F(t,M_t y)\) of Volterra type and to discontinuous problems \(\dot y(t) = f(t,M_t y)\) in the sense of A.F. Filippov. A difference method is applied to the discontinuous delay equation \(\ddot x(t) + 2D\dot x(t) + \omega ^2 x(t) = = - \operatorname{sgn} (x(t - \tau ) + \dot x(t - \tau ))\) . In the limit τ→0 we obtain results for the problem \(\ddot x + 2D\dot x + \omega ^2 x = = - \operatorname{sgn} (x + \dot x)\) which cannot be solved classically everywhere.