November 2nd, 2008, 09:46 PM  #1 
Member Joined: Sep 2008 Posts: 46 Thanks: 0  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 nonnegative x. 
November 2nd, 2008, 11:51 PM  #2 
Senior Member Joined: Jul 2008 Posts: 144 Thanks: 0  Re: differentiable function
just try to use GronwallBelman inequation. [a series of famous inequations,called Gromwall/Bellman inequation] 
November 3rd, 2008, 08:00 AM  #3 
Member Joined: Sep 2008 Posts: 46 Thanks: 0  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.).

November 4th, 2008, 02:18 AM  #4 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7  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 CauchyLipschitz theorem).

November 7th, 2008, 12:39 AM  #5 
Member Joined: Sep 2008 Posts: 46 Thanks: 0  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 integralversion, 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. 
November 8th, 2008, 04:50 AM  #6 
Senior Member Joined: Jul 2008 Posts: 144 Thanks: 0  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:Lx<=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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