Global Existence and Uniqueness for Lipschitz Ordinary Differential Equations in Integral Form

Let $a<b$ be real numbers, let $k\ge1$ be a natural number, let $\xi\in\mathbb{R}^{k}$ (Euclidean space), and let $F:[a,b]\times\mathbb{R}^{k}\to\mathbb{R}^{k}$ be a function such that the following hold, continuity of a map defined on $[a,b]$ being understood as continuity of a map of metric spaces, with $[a,b]$ regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric:

(i) (composition continuity) for every function $h:[a,b]\to\mathbb{R}^{k}$ with continuous components, the function $t\mapsto F(t,h(t))$ has continuous components;

(ii) (global Lipschitz condition) there is a real $L\ge0$ such that, with the Euclidean distance $d$,

Then there is exactly one function $x:[a,b]\to\mathbb{R}^{k}$ with continuous components such that, componentwise with the Riemann integral (whose integrands are continuous by (i), so the integrals exist by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval; degenerate intervals follow the convention of Mean-Square Riemann Integral of a Family of Random Variables),

Here exactly one means: such an $x$ exists, and any two such functions are equal at every point of $[a,b]$.