Bound of the difference between ODE and discretized system?

From: Fan Yang (yang_at_cae.wisc.edu)
Date: 11/20/04 

Date: 20 Nov 2004 05:45:00 -0500

Dear all,

First we have an ODE system

dx/dt = f(x), x belongs to R^n. There exists a unique
globally asymptotically stable fixed point.

If we do a smooth approximation by a discretized system,

x(t+1) = (1-b(t))*x(t) + b(t)*f(x(t)), where b(t) is small.

Then within a finite time, can we give an upper bound of
the distance between ODE and the discretized system above?
I was told that this distance should go to zero if the b(t) is
very small everywhere. Can somebody give me a reference
of this statement? Btw, if we take the form b(t) = 1/t, can
we get an upper bound of the difference?

Thanks a lot for your help,

Fan