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Differential equations with discontinuous right-hand sides. (English) Zbl 0724.34017
The initial value problem for an n-dimensional autonomous differential equation (1) $\dot x=f(x)$, $x(0)=x\sb 0$ with a discontinuous right-hand side is considered. There are compared the absolutely continuous functions satisfying (1) a.e. and the solutions of the corresponding differential inclusion regularization $\dot x\in F(x)$, $x(0)=x\sb 0$. In case of f with a countable set of discontinuity points $\sigma$ the author proves that the two families of solutions coincide iff $x\in \sigma$ and $0\in F(x)$ implies $f(x)=0$. More results are proved in the one-dimensional case, some examples explain the results and there is a discussion on nonautonomous systems.

34A60Differential inclusions
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
Full Text: DOI
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