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I wonder if the solution of scalar non-autonomous system

$\dot x(t)=F(x,t),\quad x(0)=x_0$

has a limit if $x(t)$ is uniformly bounded in time $t$.

Here $F$ is a Lipschitz function.

Can we say $x(t)$ is monotone after some point $t_*$?

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    @copper.hat I don't understand. Could you explain more details? – 2017-02-28
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    My apologies, I didn't read correctly, I didn't realise that it was not autonomous. – 2017-02-28

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Take the system $\dot{x}(t) = \sin t$. The solution $x(t) = x(0) - (\cos t -1) $ is bounded, but $x$ is not monotonic.

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    Thank you for your answer! I am considering an equation $\dot x(t)=a(t)-b(t)x(t)$ for any bounded function $a$ and $b$ but the solution graph always seem to converge, so I asked this question. – 2017-02-28