Talk:Bendixson–Dulac theorem

I have previously seen this stated as requiring it have the same sign across the region, rather than not being equal to zero. Which is correct?

Well, the proof linked to in the article only works if it has the same sign almost everywhere. As stated here it is absolutely false. Counterexample: take f = -y, g = x, R = x+y then (Rf)_x + (Rg)_y = - y + x, which is \neq 0 except for a set of measure 0 ( x = y), but of course x = cos(t), y = sin(t) is a nice periodic orbit. Micheal Hardy is essentially correct, it would hold if it is the same (non-zero) sign everywhere, except for a set of measure 0 (it could even be of a different sign, but only on a measure zero set. Continuity would prohibit this of course...) Mathjj (talk) 14:00, 10 May 2010 (UTC)

It would also be nice to include a section on the Bendixson-Dulac theorem in multiple dimensions —Preceding unsigned comment added by Quantum7 (talk • contribs)

I suspect a continuity assumption should be there. A continuous real-valued function on a connected region would have the same sign everywhere in the region if it is never 0. Then there's the question of having the same sign is satisfied if it's ≥0 everywhere, as opposed to being satisfied only if the inequality is strict. Michael Hardy (talk) 00:10, 29 November 2009 (UTC)