On Carathéodory’s conditions for the initial value problem

Biles, D.; Binding, P.

doi:10.1090/S0002-9939-97-03942-7

On Carathéodory’s conditions for the initial value problem
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by D. C. Biles and P. A. Binding PDF

Proc. Amer. Math. Soc. 125 (1997), 1371-1376 Request permission

Abstract:

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 have no “downward discontinuities.”

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