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.

[edit] Kneser theorem

Given an ordinary linear homogenous differential equation of the form

y'' + q (x) y = 0

with

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

continuous and q(x) > 0, then the equation is non-oscillating if

$\limsup_{x \to +\infty} x^2 q(x) < \frac{1}{4}$

and oscillating if

$\limsup_{x \to +\infty} x^2 q(x) > \frac{1}{4}.$

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