So I have the differential system :
with some initial conditions, and where .
(It's taken from the Lotka-Volterra system)
Now, I have to show that it has a unique solution. In order to prove this, we've been taught to use the Picard-Lindelöf theorem.
So I have to prove that is Lipschitz continuous in its second variable,
So far... :
I've written , using the absolute value norm. ( )
Basically, I have to prove that there exists a constant K such that :
I arrived at :
But now, I'm completely stuck... I don't see how to deal with
I'm currently looking at the mean value theorem for vector-valued functions
But my intuition tells me it's not the right way...
Then there is another question...
Show that if and are both positive, then and are both positive.
So how can I do this one ? oO Thought of finding the minimum (at (1,1) or something like that), but we wouldn't use the initial conditions ?
Any help will be appreciated Thanks !