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November 2nd, 2008, 09:46 PM   #1
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differentiable function

Let f:R --> R be differentiable where f'(x) <= a*f(x) with some constant a.

WTS that f(x) <= f(0) * e^(ax)for all non-negative x.
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November 2nd, 2008, 11:51 PM   #2
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Re: differentiable function

just try to use Gronwall-Belman inequation.
[a series of famous inequations,called Gromwall/Bellman inequation]
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November 3rd, 2008, 08:00 AM   #3
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Re: differentiable function

I tried looking this up but couldn't find much on it. Any resources. All I have are basic analysis texts (rudin,royden,etc.).
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November 4th, 2008, 02:18 AM   #4
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Re: differentiable function

Try looking for Gronwall's lemma on Google (this is under this name that this proposition is usually called; it is used in ODE theory in order to prove the uniqueness part of various existence theorems, for instance the Cauchy-Lipschitz theorem).
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November 7th, 2008, 12:39 AM   #5
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Re: differentiable function

Can you perhaps explain how to apply Gronwall's inequality to this problem. I read some references on the inequality but most of them include the integral-version, and I can't seem to apply it properly.

I think I need to come up with some function g in terms of f such that g' < 0.

still not sure.
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November 8th, 2008, 04:50 AM   #6
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Re: differentiable function

f'(x) <= a*f(x) ==> f(x)<=a*int(f,[0,x])+f(0)
then use the inequation.

now,let me show a problem:
complete lattics(L,<=) denote G={x:L|x<=f(x)},here f is a increase function on L->L
prove:if fix(f)={m},then m=supG(or,m=supG=>f(m)=m:A)
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