In mathematics, the Kneser theorem can refer to two distinct theorems in the field of ordinary differential equations: 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[1] that the equation is non-oscillating if and oscillating if 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 (non-oscillating) or negative (oscillating) because To find the solutions for this choice of
, and verify the theorem for this example, substitute the 'Ansatz' which gives This means that (for non-zero
) the general solution is where
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.
There are many extensions to this result, such as the Gesztesy–Ünal criterion.
[2] While Peano's existence theorem guarantees the existence of solutions of certain initial values problems with continuous right hand side, H. Kneser's theorem deals with the topology of the set of those solutions.
Precisely, H. Kneser's theorem states the following:[3][4] Let
be a continuous function on the region
Given a real number
, define the set
as the set of points
is a closed and connected set.