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) $L \cdot \max(t_{\max} - t_0,\ t_0 - t_{\min}) \le a$.