Imagine the following situation:
Let \begin{cases}x'(t)=f(t,x) \\x(t_0)=x_0\end{cases} be an IVP, where $f:\mathbb{R}\times \mathbb{R^n}\to \mathbb{R^n}$ is a continuous function, and $\ f$ verifies that $$||f(t,x_1)-f(t,x_2)||\leq L(t)||x_1 -x_2||,$$ with $\ L(t)$ a continuous function.
Can we prove that there's uniqueness of solution defined for all $t\in \mathbb{R}$?
I'm thinking on building a nested sequence if compact intervals and then apply Wintner's lemma to prove that we can extent the solution for all $t$. Am I right?
Thanks for your time.
$L(t)$ is a continuos function, then $\forall a,b \in \mathbb{R}, a<b,$ there's a $k$ such that $L(t) \le k$ $ \forall t \in [a,b]$
Because a continuous function reaches the maximum in a compact
Then f is locally Lipschitz, so by the Picard's Theorem the solution is unique, and use Zorn to define it in $\mathbb{R}$