Ordinary differential equations Proposition

C^1 vector fields satisfy the Picard–Lindelöf hypotheses

Builds on Picard–Lindelöf hypotheses Lipschitz condition

Statement

Let $U$ be an open subset of $\mathbb{R}^n$ , let $f : U \to \mathbb{R}^n$ be continuously differentiable, let $x_0 \in U$ and let $t_0 \in \mathbb{R}$ . Then there are real numbers $\varepsilon > 0$ , $a > 0$ and $L, K \ge 0$ with these properties. The closed ball $B$ of radius $a$ around $x_0$ lies in $U$ . The time-independent field $(t, x) \mapsto f(x)$ , extended in any way outside $B$ , satisfies the Picard–Lindelöf hypotheses at $(t_0, x_0)$ on $[t_0 - \varepsilon, t_0 + \varepsilon]$ with constants $a, L, K$ .

Remarks

Proof: choose $a > 0$ with $B \subseteq U$ . The continuous functions $\|f\|$ and $\|f'\|$ are bounded on the compact ball $B$ , by $L$ and $K$ . The mean value inequality makes $f$ $K$ -Lipschitz on $B$ . Then choose $\varepsilon$ with $L \varepsilon \le a$ . So the initial value problem $x' = f(x)$ , $x(t_0) = x_0$ has a unique local solution for each $C^1$ field $f$ . The same argument works for a time-dependent field $f(t, x)$ of class $C^1$ .

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Lipschitz condition definition
Solution of an ordinary differential equation definition
Initial value problem definition
Picard–Lindelöf hypotheses definition
C^1 vector fields satisfy the Picard–Lindelöf hypotheses statement

Tag	`03Z9V0Y78A`
URI	`m2://03Z9V0Y78A`
Kind	statement, proposition
Writing	W0 stub
MSC	34A12
Status	This copy of the site has no build data. Run `m2 status 03Z9V0Y78A`.
Uses	Picard–Lindelöf hypotheses, Lipschitz condition

C^1 vector fields satisfy the Picard–Lindelöf hypotheses

Statement

Remarks

Reading path

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