Carathéodory's existence theorem

Remarks

Such a function $x$ is called a solution in the sense of Carathéodory; it is the same as a continuous solution of the integral equation $x(t) = x_0 + \int_{t_0}^{t} f(s, x(s))\, ds$. One can take any $\beta$ with $\int_{t_0 - \beta}^{t_0 + \beta} m(s)\, ds \le b$. The proof follows that of the Peano existence theorem: one builds approximate solutions with delayed arguments. The bound by $m$ makes them equicontinuous, and the dominated convergence theorem allows passage to the limit. Example: let $f(t, x) = 0$ for $t < 0$ and $f(t, x) = 1$ for $t \ge 0$, with $x(0) = 0$. The function $x(t) = \max(t, 0)$ is a Carathéodory solution. There is no solution that is differentiable at every point, because a derivative cannot have a jump discontinuity, by Darboux's theorem. So the theorem covers equations with discontinuous time dependence, where the classical theory gives nothing.

Going deeper

For a student

The theorem replaces continuity in $t$ by measurability, and boundedness of $f$ by an integrable bound. Solutions are absolutely continuous functions, which is the natural class for the fundamental theorem of calculus of the Lebesgue integral. This framework underlies control theory, where controls are measurable functions of time. As for Peano's theorem, there is no uniqueness in general; a Lipschitz condition in $x$ with an integrable Lipschitz constant gives uniqueness, as in the Picard–Lindelöf theorem.