**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.