Gaining some insight about Picard–Lindelöf theorem. In class I have been introduced to the Picard–Lindelöf theorem. It was written down in all its technical glory.Now: the important things to remember from it were that if a function is continuous; and, suffices Lipschitz-continuity, then there is a unique solution to the given differential equation for that function. This is all very technical and surely not meant for me to be understood at this stage, but still i'd like to maybe gain some intuition behind this, especially the Lipschitz-continuity. 
 A: If $y_1$ and $y_2$ are two different solutions of some ODE $y'=f(t,y)$ with initial conditions $y(o)=y_0$, then the difference in the Picard integral equation is
$$
y_2(t)-y_1(t)=\int_0^t (f(s,y_2(s))-f(s,y_1(s)))\,ds
$$
Now one would like to have an estimate on the difference of the function values that goes to zero if the difference of the arguments goes to zero. The most simple case is Lipschitz continuity. The vector version of the mean value theorem boils down to local Lipschitz continuity, and Hölder continuity is not enough.

With Lipschitz continuity there is some constant $L>0$ such that
$$
\|f(s,y_2)-f(s,y_1)\|\le L·\|y_2-y_1\|,
$$
Using that, the difference of two solutions $y_1,y_2$ of the ODE satisfies the following integral inequality as a consequence of the Picard integral equation
$$
\|y_2(t)-y_1(t)\|\le \|y_2(0)-y_1(0)\|+L·\int_0^t \|y_2(s)-y_1(s)\|\,ds.
$$
By the Gronwall lemma this resolves to the upper bound
$$
\|y_2(t)-y_1(t)\|\le \|y_2(0)-y_1(0)\|·e^{L·|t|}
$$
which governs the sensitivity of the solutions wrt. the initial conditions. For identical initial conditions the solutions have to be identical.

With Hölder continuity, that is, there are constants $C>0$ and $0<α<1$ s.t.
$$
\|f(s,y_2)-f(s,y_1)\|\le C·\|y_2-y_1\|^α,
$$
one gets a la Gronwall as best estimate
$$
\|y_2(t)-y_1(t)\|\le \Bigl(\|y_2(0)-y_1(0)\|^{1-α}+C·(1-α)·t\Bigr)^{\frac1{1-α}}
$$
so that for identical initial conditions the right side allows some spread between the solutions. The standard examples confirm that this spread actually occurs.
A: The goal is simple: we want to prove that a unique solution exists to an ODE: 
$$
y^\prime(t)= f(t,y(t)),\quad y(t_0) = y_0
$$ We do the naive thing and use the fundamental theorem of calculus:
$$
y(t) = y(t_0) + \int_{t_0}^tf(s,y(s))ds
$$ This gives an implicit definition of $y(t)$.   We would like to use a fixed point iteration to get a unique value for $y(t)$, and the Lipschitz condition arises because of this - it's exactly what we need to prove that, on a small enough time interval, the mapping defined by the FTOC is a contraction.  If $f$ was not Lipschitz, the sequence of iterates might diverge, which isn't helpful.  It doesn't mean that a solution doesn't exist, in fact, as long as $f$ is continuous a solution always exists, it's just that we can't prove that it exists (and is unique) using the contraction mapping technique.
A: The theorem roughly states: an initial value problem to an ODE (in explicit form, that is, $y'(t)=f(t,y(t))$) has a unique solution for sufficiently well-behaved $f$. The question is what exactly "sufficiently well-behaved" means, and to name a criterion that is useful in practice, it works if $f$ is in $C^1$. The local Lipschitz condition is a weaker condition that is more technical, so the formulation of the PL theorem with the Lipschitz condition instead of the $C^1$ condition is better in the sense that the theorem is stronger, but is not necessarily better for didactic purposes. And how far this trading generality for simplicity goes depends on authors/lecturers: for example Lipschitz continuity of $f$ is not required for the $t$ variable (making this precise is even more technical) and some authors formulate the theorem in this way, some don't. But the most useful first approximation to remember the theorem is: it definitely works for $f$ in $C^1$, and for other $f$ maybe too.
