Local Lipschitz continuity in the state variable, locally uniform in time and parameters

Let $n\in \mathbb{N}$ with $n\ge 1$, let $D\subseteq \mathbb{R}\times {\mathbb{R}}^n$ be open, and let $F:D\to {\mathbb{R}}^n$. The map $F$ is locally Lipschitz in the state variable, locally uniformly in time, if every compact time-state cylinder $C\subseteq D$ has a finite $L\ge 0$ such that

$\mathrm{\parallel }F(t,x)-F(t,y){\mathrm{\parallel }}_2\le L\mathrm{\parallel }x-y{\mathrm{\parallel }}_2$

whenever $(t,x),(t,y)\in C$ have the same time coordinate.

On every compact time-state cylinder the state-variable inequality holds with one finite constant $L$. More generally, let $p\ge 1$, let $\mathrm{\Lambda }\subseteq {\mathbb{R}}^p$, let $D_{\mathrm{\Lambda }}\subseteq \mathbb{R}\times {\mathbb{R}}^n\times \mathrm{\Lambda }$ be open in its relative Euclidean topology, and let $F:D_{\mathrm{\Lambda }}\to {\mathbb{R}}^n$. The parameter family is locally Lipschitz in the state variable, locally uniform in time and parameters when every compact time-state-parameter cylinder $C\subseteq D_{\mathrm{\Lambda }}$ has one finite $L\ge 0$ such that

$\mathrm{\parallel }F(t,x,\lambda )-F(t,y,\lambda ){\mathrm{\parallel }}_2\le L\mathrm{\parallel }x-y{\mathrm{\parallel }}_2$

whenever $(t,x,\lambda ),(t,y,\lambda )\in C$ have the same time and parameter coordinates.