I am studying the uniqueness of solution for some ODE that showed in my research. For that I want to use Gronwall's lemma, but it all boils down to finding an upper bound $C$:
$$ \lvert ax - by \rvert \leq C \lvert x - y \rvert$$
Where $x,y$ are both strictly positive (and bounded actually) and $a,b \in [\epsilon, M]$ for some positive $\epsilon$.
So far, I have narrowed the problem to finding a bound for
$$ \lvert ax - by \rvert = \lvert a(x-y)+y(a-b) \rvert $$
but I really need the $(x-y)$ factor. Any ideas on how to move forward (or of an alternative approach) are welcome