On Carathéodory’s conditions for the initial value problem. (English) Zbl 0869.34004
Summary: We prove a local existence theorem of Carathéodory-Goodman type for \(\dot x(t)= f(t,x(t))\) where instead of \(f(t,\alpha)\) being continuous in \(\alpha\) we require only that it has no “downward discontinuities”.
MSC:
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 34A40 | Differential inequalities involving functions of a single real variable |
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