bibliography for weak solutions of ODE's Could someone recommend to me some bibliography about weak solutions of ODE's and solutions of ODE's that are not Lipschitz or discontinuous??
 A: Discontinuous ODE may be called "ODE with discontinuous right hand side" or "Filippov's systems". They where studied in the 70's or 80's by A. F. Filippov who published a book on them titled "Differential Equations with Discontinuous Righthand Sides". Such topic was exploited by V. Utkin to develop the so-called Sliding Mode Control. In such a control technique, the discontinuity of the control action provide robustness to the control system. There is lot of literature about that. 
As literature about discontinuous ODEs I can recommend "Discontinuous Systems by Orlov, Y." and the first two chapters of the book by Filippov.
An interesting approach my be found in "Hybrid Dynamical Systems" by Goebel R., Sanfelice, R. G. & Teel, A. R.. There, such kind of discontinuous differential equations are treated as hybrid systems, where a discrete part interacts with a continuous part. This approach based in hybrid systems is very popular.
Regarding numerical solutions to such systems, I strongly recommend the paper "Sliding Motion in Filippov Differential Systems: Theoretical Results and a Computational Approach by Dieci, L. & Lopez, L."
