Kneser theorem

In mathematics, in the field of ordinary differential equations, the Kneser theorem, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not.

Statement of the theorem

Consider an ordinary linear homogenous differential equation of the form

$-y\text{'}\text{'}\; +\; q(x)y\; =\; 0$

with

$q:\; [0,+infty]\; to\; mathbb\{R\}$

continuous and q(x) > 0.
We say this equation is oscillating if it has a solution y with infinitely many zeros, and non-oscillating otherwise.
The theorem states that the equation is non-oscillating if

and oscillating if

$limsup\_\{x\; to\; +infty\}\; x^2\; q(x)\; >\; -frac\{1\}\{4\}.$