Kneser's theorem (differential equations)


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 homogeneous differential equation of the form
with
continuous.
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

Example

To illustrate the theorem consider
where is real and non-zero. According to the theorem, solutions will be oscillating or not depending on whether is positive or negative because
To find the solutions for this choice of, and verify the theorem for this example, substitute the 'Ansatz'
which gives
This means that the general solution is
where and are arbitrary constants.
It is not hard to see that for positive the solutions do not oscillate while for negative the identity

shows that they do.
The general result follows from this example by the Sturm–Picone comparison theorem.

Extensions

There are many extensions to this result. For a recent account see.