Picard–Lindelöf hypotheses

Definition

Let $t_{\min} \le t_0 \le t_{\max}$ be real numbers, let $J = [t_{\min}, t_{\max}]$, let $x_0 \in \mathbb{R}^n$, and let $a, L, K \ge 0$. Let $B = \{x \in \mathbb{R}^n : \|x - x_0\| \le a\}$, and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ be a function. The function $f$ satisfies the Picard–Lindelöf hypotheses at $(t_0, x_0)$ with constants $a, L, K$ if these four conditions hold. (1) For each $t \in J$, the map $x \mapsto f(t, x)$ is $K$-Lipschitz on $B$ (Lipschitz condition). (2) For each $x \in B$, the map $t \mapsto f(t, x)$ is continuous on $J$. (3) For each $t \in J$ and $x \in B$, $\|f(t, x)\| \le L$. (4) $L \cdot \max(t_{\max} - t_0,\ t_0 - t_{\min}) \le a$.

Remarks

Conditions (1) and (2) together imply that $f$ is continuous on $J \times B$. Condition (4) makes the time interval short enough. A curve that starts at $x_0$ with speed at most $L$ cannot leave $B$ during the time interval $J$. Under these hypotheses, the Picard–Lindelöf theorem gives a unique solution of $x' = f(t, x)$, $x(t_0) = x_0$ on $J$. A $C^1$ vector field satisfies these hypotheses on a small time interval.

Explanations

L1 human

These are the conditions that make the proof of existence and uniqueness work. The field must be continuous in time, Lipschitz in space and bounded near the start point. The time interval must be so short that the solution stays in the region where we control the field.