Global Solutions of Ordinary Differential Equations Background
Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying,


*

*$f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$,

*for every compact $K \subseteq {\mathbb R}^n$ and every $b > a \geqslant 0$, $$
\int_a^b \|f(t,\cdot)\|_K\,dt < \infty\,,
$$ 
where
$$
\|f(t,\cdot)\|_K := \sup_{x \in K} |f(t,x)| + \sup_{\substack{x,y \in K \\ x \neq y}} \frac{|f(x) - f(y)|}{|x - y|}\,,
$$
and

*there exist locally integrable $\alpha, \beta: [0, \infty) \rightarrow [0, \infty)$ such that
$$|f(t,x)| \leqslant \alpha(t)|x| + \beta(t)$$
for every $(t,x) \in [0,\infty) \times {\mathbb R}^n$.


We know that, under these hypotheses, the ordinary differential equation
$$
x' = f(t,x)
$$
generates a unique global flow
$$
\varphi: [0,\infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n\,.
$$
Question
I'm wondering whether there is a canonical way of realizing $\varphi$ as the limit of globally defined maps
$$
\varphi_k: [0,\infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n\,,
\quad k \in {\mathbb N}\,.
$$
Progress


*

*One may assume without loss of generality that $f(t,\cdot)$ is compactly supported uniformly in $t$; that is, there exists a compact $K \subseteq {\mathbb R}^n$ such that $f(t,x) = 0$ whenever $x \notin K$.

*Setting $\varphi_0(t,x) := x$, then
$$
\varphi_k(t,x) := x + \int_0^t f(s, \varphi_{k-1}(s,x))\,ds\,,
\quad
k = 1, 2, 3, \ldots\,,
$$
recursively, as in the standard proof of existence, doesn't seem to work.

*Euler's proof of existence seems to almost work. But the construction is not "canonical" in the sense that the subsequence along which convergence occurs via Arzela-Ascoli will depend on $f$. I would like to have a sequence $(\varphi_k)_{k \in {\mathbb N}}$ which could be defined by a procedure independent of $f$. This is because I'm ultimately interested in the flow of parametrized differential equations, and I don't want to have to choose a different subsequence for each parameter.

 A: I think I got it! I turned out to be mistaken about item 2 in my Progress Notes.
Lemma 1. Suppose $f \colon {\mathbb R}_{\geqslant 0} \rightarrow {\mathbb R}_{\geqslant 0}$ is locally integrable, and let $F\colon {\mathbb R}_{\geqslant 0} \rightarrow {\mathbb R}_{\geqslant 0}$ be the primitive given by
$$
F(t) := \int_0^t f(s)\,ds\,, \quad t \geqslant 0\,.
$$
For any positive integer $m$, set $f\colon {\mathbb R}_{\geqslant 0}^m \rightarrow {\mathbb R}_{\geqslant 0}$ by 
$$
f_m(t) := \prod_{i = 1}^m f(t_j)\,, \quad t = (t_1,\ldots,t_m) \in {\mathbb R}_{\geqslant 0}^m\,,
$$
and set
$$
S_m(T) := \{t \in {\mathbb R}_{\geqslant 0}^m\,;\ 0 \leqslant t_1 \leqslant \cdots \leqslant t_m \leqslant T\}\,,
\quad
T \geqslant 0\,.
$$
Then
$$
\int_{S_m(T)} f_m(t)\,dt = \frac{[F(T)]^m}{m!}\,,
\quad
\forall T \geqslant 0\,,
\quad
\forall m = 1, 2, 3, \ldots\,.
$$
Proof. Follows by Fubini and integration by substitution. (Details/clarification upon request.)
Assuming that $f(t,\cdot)$ is compactly supported uniformly in $t$ as described in item 1 of my Progress Notes, we may apply Lemma 1 to show that $(\varphi_k(\cdot,x))_{k \geqslant 0}$ is a Cauchy sequence on $[0,T]$ for each $T \geqslant 0$. Thus
$$
\varphi(t,x) = \lim_{k \to \infty} \varphi_k(t,x)\,,
\quad
\forall (t,x) \in {\mathbb R}_{\geqslant 0} \times {\mathbb R}^n\,.
$$
