[Updated: 17-5-2011 for a natural corollary. ] Last week, when I was giving a seminar I encountered a problem in analysis (which I was unable to solve at that time). After discussing with John Ma, we have come up with the following: (we write in place of the more clumsy and instead of . )
Then for any , there exists such that
As a corollary, one can easily see that we can relax our conditions that nowhere zero to the conditions that not identically zero.
For example, can we find such such that
to be any arbitrary point in ? This theorem tells us that the answer is yes.
Then for any , there exists , such that
Proof: First note that can be made exact up to an integrating factor, i.e.
for some nowhere zero function . In fact, take to be the integrating factor
which is well-defined and nowhere zero. Then . Thus we have
where . There is no need to explicitly write out all terms of , except to note that
for some continuous functions with nowhere zero on .
(Such can be obtained by, for example, prescribing the initial values , but this is not very important here. ) As is nozero, we can find such that to be (first make this nonzero, then rescale ). But then by (1) and (2), we have
By prescribing the initial values , such must be non-constant. In particular, there exists such that is well-defined and strictly decreasing on . We then take nonzero odd function such that for . Now we take on which can then be extended trivially on . We thus have
as is odd and is strictly decreasing. By rescaling , we have
This completes the proof.
Similarly, we define
Then by the lemma, there exists , such that for
We now extend trivially to and extend to such that . (By definition a function on a compact set is one which can be extended to be a function on a slightly bigger open set. ) We claim that the required is then
In fact, as is supported in ,
This completes the proof of John Ma’s theorem.