Picard's existence theorem, successive approximations and the global solution

First I prove a lemma :

If $F$ has one iteration which is a contraction, then $F$ has a unique fixed point.

Proof : If $F^n$ is a contraction then it has a unique fixed point $\psi$. The equalities $$F^n\circ F(\psi)=F\circ F^n(\psi)=F(\psi)$$ show that, by unicity of fixed point, $F(\psi)=\psi,$ and so existence of fixed point for $F.$ For unicity, we see that the existence a new fixed point $\varphi$ implies that $$F^n(\varphi)=F\circ\dots\circ F\circ F(\varphi)=F\circ\dots\circ F(\varphi)=\dots=\varphi$$ and so $\varphi=\psi.$ $\square$

Now let $r\in\mathbb{R}^+,$ and let's show that one of the iterated of $F$ is a contraction on $C([0,r]).$ We have that : $$|(F(\psi)-F(\varphi))(t)|=|\int_0^t\psi(s)-\varphi(s)|\mathrm{d}s\leq t\,||\psi-\varphi||_\infty$$ and so $$||F(\psi)-F(\varphi)||_{\infty}\leq r\cdot||\psi-\varphi||_{\infty}$$ so $F$ is $r-$lipschitz. Now see that $$|(F^2(\psi)-F^2(\varphi))(t)|\leq\int_0^t|F(\psi(s))-F(\varphi(s))|\mathrm{d}s\leq\int_0^t s||\psi-\varphi||_\infty\mathrm{d}s=\frac{t^2}{2}||\psi-\varphi||_{\infty}.$$

One can see that $$||F^n(\psi)-F^n(\varphi)||_\infty\leq\frac{r^n}{n!}||\psi-\varphi||_\infty$$ and as it exists $n\in\mathbb{N}$ such that $n!>r^n$ we establish our result.

The global solution of an initial value problem (IVP) $$x'=f(t,x),\quad x(t_0)=x_0\tag{1}$$ is not obtained via Picard's theorem, but through analytic continuation. The grand picture is as follows:

We are given an open domain $\Omega\subset{\mathbb R}\times{\mathbb R}^n$ and a continuous function $$f:\quad \Omega\to{\mathbb R}^n,\qquad(t,x)\mapsto f(t,x)$$ ($n=1$ in the sequel) which is locally Lipschitz continuous with respect to the second variable. This means that for every point $(t_0,x_0)\in\Omega$ there is a rectangular window $W$ with center $(t_0,x_0)$ and a constant $C$ such that $$|f(t,x)-f(t,x')|\leq C|x-x'|\qquad\bigl((t,x),(t,x')\in W\bigr)\ .$$ An $f\in C^1(\Omega)$ automatically fulfills these conditions. Picard's theorem then guarantees that for every $(t_0,x_0)$ there is a window $$[t_0-h,t_0+h]\times [x_0-q,x_0+q]\subset W$$ and a micro-solution $$\phi_0:\quad[t_0-h,t_0+h]\to [x_0-q,x_0+q] \tag{2}$$ of $(1)$.

The micro-solution $(2)$ is continuous up to $t_1:=t_0+h$, and even satisfies $\phi_0'(t_1-)=f\bigl(t_1,\phi_0(t_1)\bigr)$. The uniqueness part of Picard's theorem then allows to conclude that immediately to the left of $t_1$ this $\phi_0$ coincides with the micro-solution $\phi_1$ of the IVP $$x'=f(t,x),\quad x(t_1)=\phi_0(t_1)\ .$$ Note that $\phi_1$ is defined up to the point $t_2:=t_1+h'$ for some $h'>0$, hence extends $\phi_0$ further to the right. Concatenating $\phi_0$ and $\phi_1$, and proceeding in this way forever (and similarly to the left of $t_0$) we arrive at the global solution $$\phi_*:\quad I_* \to{\mathbb R}$$ of the IVP $(1)$. The interval $I_*$ on which $\phi_*$ is defined may be infinite, and in general depends on the initial point $(t_0,x_0)$. About this $\phi_*$ one can say the following:

(i) Global uniqueness: Any solution $\phi$ of $(1)$, defined in some open interval $I\subset{\mathbb R}$, is part of $\phi_*$. This means that $I\subset I_*$, and that $\phi(t)=\phi_*(t)$ for all $t\in I$.

(ii) The graph of $\phi_*$ extends to infinity, or to the boundary of $\Omega$. More precisely: Given any compact set $K\subset\Omega$ there are points $\bigl(t,\phi_*(t)\bigr)\notin K$.

These things are proven in detail in books titled "Theory of differential equations".

For ODE defined on a stripe $(a,b)×\Bbb R^n$ with a global Lipschitz constant $L$ there, one can use the modified supremum norm $$ \|y\|_L=\sup e^{-2L·|t-t_0|}\|y(t)\| $$ where the initial condition is at $t_0\in (a,b)$. One can then show that in this norm the Picard iteration is contractive for continuous functions over the whole of $(a,b)$, giving the maximally possible interval at once.