Ordinary differential equations Existence, uniqueness and dependence on data Theorem

Peano existence theorem

Builds on Initial value problem Solution of an ordinary differential equation

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.

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Solution of an ordinary differential equation definition
Initial value problem definition
Peano existence theorem statement

Where this leads

3 pages build on this one.

Tag	`098YP831QY`
URI	`m2://098YP831QY`
Aliases	`cauchy-peano-theorem`, `peano-theorem`
Kind	statement, theorem
Writing	W1 stated
MSC	34A12
Status	This copy of the site has no build data. Run `m2 status 098YP831QY`.
Uses	Initial value problem, Solution of an ordinary differential equation
Same as	Wikidata Q740093 Wikipedia Peano existence theorem

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