For the purposes of differential geometry I would like to fully process the basics of ODEs, namely the existence/uniqueness theorem (EUT) and continuous/smooth dependence on initial conditions and parameters (CSD), in maximum generality. Many standard ODE textbooks relegate the general EUT to the back (assuming they cover it at all, which isn't always) and focus on linear systems or explicit examples, and most don't even mention parameters or CSD.
- Firstly, I'd like to clarify what the right definitions are, because almost every text has its own conventions: my current reference, Bröcker's "Analysis in mehreren Variablen", defines an ODE with parameter as a continuous function$$f:I\times U\times V\to\Bbb R^n$$with $I$ a non-empty open interval, $U\subset\Bbb R^n$ open, and $V\subset\Bbb R^k$ (nothing is stated about it, so I assume it's an arbitrary euclidean set), while a solution to parameter $v\in V$ and initial condition $(t_0,x_0)\in I\times U$ is a differentiable function $x:J\to U$ for $J\subset I$ a non-empty open interval such that $$x'(t)=f(t,x(t),v),\qquad x(t_0)=x_0.$$Is this definition correct? The text goes on to say that in general $V$ may be an arbitrary metric space without it affecting anything, but then in the proof of the existence theorem it begins by saying that "after shrinking $I,U,V$ we may assume $f$ is bounded", which would require $V$ to be locally compact, thus ruling out many metric spaces (and euclidean sets). What is the correct framework here? How would one even formulate smooth dependence in the case of general metric spaces?
- What is an adequate reference for a meticulous, non-handwavy proof of CSD? So far besides Bröcker (which has many typos and, possibly, inconsistencies) I only have Lee's "Introduction to Smooth Manifolds" and Conlon's "Differentiable Manifolds". Both are fine, if a little complicated, but are there no alternatives? Also they don't cover equations with parameters (I know such equations can be reduced to autonomous ones, but I'd still like a general formulation). The result I'm looking for should be roughly of this shape:
If $f$ is $C^k$ for $k\geq0$, satisfying the Lipschitz condition in $x$ uniformly for $(t,v)$ (possibly locally), and $(t_0,x_0,v_0)$ is arbitrary, then there exists an open neighbourhood $(t_0,x_0,v_0)\in I_0\times U_0\times V_0$ and a $C^k$ function $\alpha:I_0\times U_0\times V_0\to\Bbb R^n$ such that for $(x,v)$ fixed $\alpha(t,x,v)$ is a solution of the equation with parameter $v$ and initial condition $(t_0,x)$. [Here I suppose we must assume $V$ is euclidean open to discuss differentiability of $\alpha$]