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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?

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Is $u$ supposed to be the same as $y$? –  Robert Israel Nov 20 '12 at 1:39
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up vote 1 down vote accepted

Assuming you meant $\dot{y} = f(t) - g(t) y$, this is a linear differential equation and has the explicit solutions $y(t) = \mu(t)^{-1} \int \mu(t) f(t)\ dt$ where $\mu(t) = \exp(\int g(t)\ dt)$, from which it is clear that you have global existence of solutions. Uniqueness follows from the standard existence and uniqueness for ODE.

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