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I am trying to prove that the fixed point of following first order ODE is stable.

$\dot x(t)=a(t)-b(t)x(t),\quad x(0)=x_0>0$

We know that $x(t)$ is nonnegative and uniformly bounded above a priori.

Also, we have the following property (*)

(*): $a(t),b(t) \in A \subset (0,\infty),~\forall t$ for some bounded interval $A$.

I want to prove that $\lim_{t \to \infty}\dot x(t)=0$.

I tried to show that $\lim_{t \to \infty}x(t)$ exists and apply Barbalat's lemma.

I want to get rid of property (*) but i'm assuming it at this stage. Thank you in advance.

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    Shift the equation by using $u(t)=e^{Mt}y(t)$ where $M=\sup A$. Then apply Grönwall lemma, possibly also using $m=\inf A>0$. – 2017-02-27
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    Or just use the estimates in the solution formula, or use $\dot x\le M-mx$ using the bounds above. – 2017-02-27
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    Could you give more detail?all i can find is just upper and lower bound of x(t) – 2017-02-28

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