Kneser theorem

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" + q(x)y = 0with:q: [0,+infty] o 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:liminf_{x o +infty} x^2 q(x) < -frac{1}{4}and oscillating if:limsup_{x o +infty} x^2 q(x) > -frac{1}{4}.


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