Picard's existence theorem says that equation $$y'(x)=f(x, y(x)),~y(x_0)=y_0,$$ where $f$ has some properties, has a unique solution. Moreover it allows one to construct an iterative approximate solution.
I wonder whether there is a result allowing one to say anything (existence, uniqueness) about equation:
$$y'(x)=f(x, y(g(x))),~y(x_0)=y_0,$$
where $g$ and $f$ are some functions (e.g. analytic - just continuity or Lipschitz continuity may be too weak to say anything interesting).