> [!definition] > > Let $0 \in J \subset \mathbb R$ be an open interval, $E$ be a [[Banach Space|Banach space]], $U \subset E$ be [[Open Set|open]], and $f: J \times U \to E$ be a [[Time-Dependent Vector Field|time-dependent vector field]] of class $C^p$. > > An **integral curve** for $f$ with **initial condition** $x_0 \in U$ is a [$C^p$](Space%20of%20Continuously%20Differentiable%20Functions)-mapping $\alpha: J_0 \to U$ such that $0 \in J_0 \subset J$ is open, $\alpha(0) = x_0$, and $\alpha'(t) = f(t, \alpha(t))$. > [!definition] > > Let $f: J \times U \to E$ be an integral curve, then $f$ is **Lipschitz** if the mapping > $ > U \to C(J, E) \quad x \mapsto f(\cdot, x) > $ > is Lipschitz with respect to the uniform norm on $C(J, E)$. The **Lipschitz constant** of the > [!theorem] > > Let $\alpha: J_0 \to U$ such that > $ > \alpha(t) = x_0 + \int_0^t f(t, \alpha(t))dt > $ > then $\alpha \in C^p$ and is an integral curve with initial condition $x$. > > *Proof*. By the fundamental theorem of calculus, if $\alpha \in C^k$ where $k < p$, then $t \mapsto f(t, \alpha(t))$ is $C^{k}$ and $\alpha \in C^{k+1}$. By induction, $\alpha \in C^p$.