TITLE: Questions about the Picard–Lindelöf theorem for an ODE
QUESTION [19 upvotes]: In the sketch proof of Picard–Lindelöf existence theorem for an
ODE:

It can then be shown, by using the Banach fixed point theorem, that the sequence of "Picard iterates" $φ_k$ is convergent and that
  the limit is a solution to the problem. Exploiting the fact that
  the width of the interval where the local solution is defined is entirely determined by the Lipschitz constant of the function, one
  can assure global existence of the solution, i.e. the solution
  exists and is unique until it leaves the domain of definition of
  the ODE. An application of Grönwall's lemma to $|φ(t) − ψ(t)|$,
  where $φ$ and $ ψ$ are two solutions, shows that $φ(t) = ψ(t)$, thus
  proving the global uniqueness (the local uniqueness is a
  consequence of the uniqueness of the Banach fixed point).

In the same article, it latter says

The importance of this result is that the interval of definition of the solution does eventually not depend on the Lipschitz constant
  of the field, but essentially depends on the interval of
  definition of the field and its maximum absolute value of it.

Wikipedia also mentions 

... which (both Peano existence theorem and Picard–Lindelöf theorem) are both local results. ...
When the hypotheses of the Picard–Lindelöf theorem are satisfied, then local existence and uniqueness can be extended to a
  global result. More precisely:[20] For each initial condition
  $(x_0, y_0)$ there exists a unique maximum (possibly infinite) open
  interval $$
        I_{max} = (x_-,x_+), x_\pm \in \mathbb{R}, x_0 \in I_{max} $$ such that any solution which satisfies this initial condition is a
  restriction of the solution which satisfies this initial condition
  with domain $I_{max}$. ...  Note that the maximum domain of the
  solution ... may depend on the specific choice of $(x_0, y_0)$.


My questions:

Global or local existence and uniqueness: I was wondering what "local" and "global" mean when referring  to existence and/or uniqueness of the solution to an ODE?
In the second quote, the Picard–Lindelöf theorem is mentioned first as a local result, and the maximum domain $I_{max}$ of the solution both contains and depends on $(x_0, y_0)$, which is how I guess what "local" is. So why is the Picard–Lindelöf theorem said to be " a global result" in the second quote, and "global existence" and "global uniqueness" in the first quote ?
Dependence on the Lipschitz constant: In the first article, it first says "the width of the interval where the local solution is defined is entirely determined by the Lipschitz constant of the function", but latter says "the interval of definition of the solution does eventually not depend on the Lipschitz constant of the field". I was wondering if it is self-contradicting?
Proof of Uniqueness: In the first quote, it says "Exploiting the fact that the width of the interval where the local solution is defined is entirely determined by the Lipschitz constant of the function, one can assure global existence of the solution, i.e. the solution exists and is unique until it leaves the domain of definition of the ODE". Is "unique" assured here, or by an application of Grönwall's lemma mentioned in the next sentence in the quote?

Thanks and regards!

REPLY [6 votes]: There are also global variants of the Picard-Lindelöf theorem. If $f$ is defined on $I\times \Bbb R^n$ and satisfies a global Lipschitz condition in the second argument there, that is one constant $L$ for the full domain, then unique solutions $I\to\Bbb R^n$ to any IVP exist. This can be directly proven using the Banach fixed point theorem on the integral formulation of the IVP in the norm $$\|x\|_L=\sup_{t\in I}e^{-2L|t-t_0|}\|x(t)\|,$$ as in that norm the Picard iteration has contraction factor $\frac12$.

See also

a more detailed proof  or here, 
a different method for the same global result, 
or a little more general for a time-dependent Lipschitz condition.