So I was trying to do some existence and uniqueness results beyond the trivial setting. So consider the 1D non-autonomous ODE given by
$\dot{y} = f(t) - g(t) y $ where $f,g \geq 0$ are integrable and $f(t),g(t) \rightarrow 0$ for $t \rightarrow \infty$. How would I go about proving the existence and uniqueness for the solution of such an ODE for $t \rightarrow \infty$?
Just by a contraction argument?