Gronwall’s Inequality
We’re going to need another analytic lemma, this one called “Gronwall’s inequality”. If is a continuous, nonnegative function, and if
and
are nonnegative constants such that
for all then for all
in this interval we have
That is, we can conclude that grows no faster than an exponential function. Exponential growth may seem fast, but at least it doesn’t blow up to an infinite singularity in finite time, no matter what Kurzweil seems to think.
Anyway, first let’s deal with strictly positive . If we define
then by assumption we have . Differentiating, we find
, and thus
Integrating, we find
Finally we can exponentiate to find
proving Gronwall’s inequality.
If , in our hypothesis, the hypothesis is true for any
in its place, and so we see that
for any positive
, which means that
must be zero, as required by Gronwall’s inequality in this case.
[…] Gronwall’s inequality tells us that , which is exactly the inequality we asserted […]
Pingback by Control on the Divergence of Solutions « The Unapologetic Mathematician | May 13, 2011 |
[…] turns out to be a lot more messy, but essentially doable by similar methods and a generalization of Gronwall’s inequality. For the sake of getting back to differential equations I’m going to just assert that not […]
Pingback by Smooth Dependence on Initial Conditions « The Unapologetic Mathematician | May 16, 2011 |
[…] Armstrong: Another Existence Proof (of the convergence of the Picard iteration), Gronwall’s Inequality, Lie […]
Pingback by Sixth Linkfest | May 25, 2011 |