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
Extensions
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 , and such that for all .
Given a real number satisfying , define the set as the set of points for which there is a solution of such that and . The set is a closed and connected set.