In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a
has the same sign (
has no periodic solutions lying entirely within the region. "Almost everywhere" means everywhere except possibly in a set of measure 0, such as a point or line.
The theorem was first established by Swedish mathematician Ivar Bendixson in 1901 and further refined by French mathematician Henri Dulac in 1933 using Green's theorem.
Proof
Without loss of generality, let there exist a function
in simply connected region
But on