Peano existence theorem

Statement

Let $D$ be an open subset of $\mathbb{R} \times \mathbb{R}^n$, let $f : D \to \mathbb{R}^n$ be continuous, and let $(t_0, x_0) \in D$. Then there are an open interval $I$ with $t_0 \in I$ and a differentiable function $z : I \to \mathbb{R}^n$ with these properties. For each $t \in I$, the point $(t, z(t))$ is in $D$ and $z'(t) = f(t, z(t))$. Also, $z(t_0) = x_0$.

Remarks

The theorem needs only continuity of $f$. It gives existence, but not uniqueness. See the non-uniqueness example. With a Lipschitz condition in $x$, the Picard–Lindelöf theorem gives existence and uniqueness. A common proof builds Euler polygons with smaller and smaller steps and takes a uniformly convergent subsequence by the Arzelà–Ascoli theorem.

History

Giuseppe Peano published the theorem in 1886, with a proof that was not correct. In 1890 he published a correct proof that uses successive approximations. The name Cauchy–Peano theorem also honours Augustin-Louis Cauchy.

Explanations

L1 human

If the vector field is continuous, at least one solution starts at each point. The theorem does not say that the solution is unique, and it gives the solution only for a short time.