ODE fundamental theorem proof – possible error/omission in Lee

Question

I was reading John Lee's wonderful book Introduction to Smooth Manifolds and believe I found a small mistake or omission in the proof of Theorem D.6 (Fundamental Theorem for Nonautonomous ODEs) on page 672. Non-typo mistakes in this book seem to be exceedingly rare, so I wanted to check that my understanding of this proof is correct. I want to focus on just the existence part of the statement.

The equations (D.1) and (D.2) are the following.

Stated informally, $V$ is a vector field dependent on time and defined on a space-time cylinder $J\times U$. For any point $(s_0,x_0)$ within that cylinder, there is a cylindrical neighborhood $J_0\times U_0$ containing it so that we can pick initial conditions $(t_0,c)$ from $J_0\times U_0$ and find a function $y$ satisfying the ODE and defined for all time in $J_0.$ The proof of the fundamental theorem for nonautonomous ODEs relies on the fundamental theorem for autonomous ODEs. We only need the existence part.

Notice that the initial time $t_0$ is fixed before we get the domain of definition $J_0,$ in contrast to the fundamental theorem of nonautonomous ODEs where we can choose an initial time $t_0$ after fixing a domain of definition $J_0.$ The basic idea of the proof is that every non-autonomous ODE corresponds to an autonomous ODE when we make $t$ a variable. Given a non-autonomous ODE, we replace it with this autonomous ODE and apply the fundamental theorem for autonomous ODEs. In the proof, this fundamental theorem allows us to fix a time variable $s_0$ that we'll use for the initial condition of the autonomous ODE. On the other hand, we need to be able to choose the initial condition $t_0$ of the corresponding non-autonomous ODE freely in a neighborhood of $s_0$ to fulfill the existence part of Theorem D.6. This is permitted by the existence portion of the fundamental theorem on autonomous ODEs because the variable $t$ is a dependent "space" variable in the autonomous ODE.

My concrete objection to Lee's version of the proof is that the statement "Theorem D.1 guarantees that… there exists a unique solution to (D.20)" is not correct because, after fixing $s_0$, Theorem D.1 can only make statements about $y^i(s_0)$ rather than $y^i(t_0).$ Of course we could apply Theorem D.1 for $t_0$ but then the domain of definition $J_0$ on which the solution existed would be dependent on $t_0$, not fulfilling the claim of Theorem D.6.

At this point, I believe that the last two equations of (D.20) should written in terms of the variable $s$ as $\dot{y}^0(s)=1,\dot{y}^i(s)=…, y^0(s_0)=t_0$ and $y^i(s_0)=c^i$ so that we can define $t=y^0(s).$ The fundamental theorem of autonomous ODE gives a solution to the system (D.20), and since $\dot{y}_0(s)=1$ we'll get $t=y^0(s)=s+(t_0-s_0).$ At this point, to get a path satisfying the non-autonomous ODE, I think we need to do a change of variables to express the path in terms of $t$, letting $Y(t)=y(t-(t_0-s_0))$ to find a function satisfying (D.1-D.2). Then $Y(t_0)=y(s_0)=c$ as required.

Am I right?

Edit: I think a problem with my solution is that the domain on which $Y$ is defined would be $J+(t_0-s_0)$ which depends on $t_0$.

Answer 1

I think the key to explaining my question is that in the proof of Theorem D.6, Lee actually uses a result slightly stronger than Theorem D.1 which can be seen as a corollary of Theorem D.1. This slightly stronger result would replace the existence portion of D.1 with

EXISTENCE: For any $s_0\in \mathbb{R}$ and $x_0\in U$, there exist an open interval $J_0$ containing $s_0$ and an open subset $U_0\subseteq U$ containing $x_0$ such that for each $t_0\in J_0$ and $c\in U_0$, there is a $C^1$ map $y:J_0\to U$ that solves $(D.3)-(D.4)$.

This "corollary" can be proven by restricting to a smaller $J_0$ and using the fact that the system is autonomous.