Existence and Uniqueness for Ordinary Differential Equations with Measurable Time Dependence

Let $n$ be a natural number, let $T>0$, $K\ge0$ and $\Lambda\ge0$ be real numbers, and let

be a map into Euclidean space such that:

1. (Measurability in time.) For every $x\in\mathbb{R}^n$ and every $i\in\{1,\dots,n\}$ the map $t\mapsto f^i(t,x)$ is measurable with respect to the trace Borel $\sigma$-algebra on $[0,T]$ and the Borel $\sigma$-algebra on the real line.

2. (Boundedness.) $|f(t,x)|\le K$ for all $t\in[0,T]$ and $x\in\mathbb{R}^n$, where $|\cdot|$ is the Euclidean norm.

3. (Lipschitz dependence on the state.) $|f(t,x)-f(t,y)|\le\Lambda|x-y|$ for all $t\in[0,T]$ and $x,y\in\mathbb{R}^n$; that is, $f(t,\cdot)$ is Lipschitz with constant $\Lambda$ for each $t$.

Then the following hold.

(a) (Integrands along continuous paths.) For every continuous map $y:[0,T]\to\mathbb{R}^n$ and every $i$, the map $s\mapsto f^i(s,y_s)$ is measurable on $[0,T]$ and bounded in absolute value by $K$, so its Lebesgue integral over $[0,t]$ exists for every $t\in[0,T]$.

(b) (Existence and uniqueness.) For every $x_0\in\mathbb{R}^n$ there is exactly one continuous map $x:[0,T]\to\mathbb{R}^n$, written $t\mapsto x_t$, such that

where the integral over $[0,0]$ is $0$; in particular its value at $t=0$ is $x_0$.

(c) (Regularity.) This solution satisfies $|x_t-x_r|\le K|t-r|$ for all $r,t\in[0,T]$, so it is Lipschitz with constant $K$.