In mathematics, the Kneser theorem can refer to two distinct theorems in the field of ordinary differential equations:

Statement of the theorem due to A. Kneser

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 a is real and non-zero. According to the theorem, solutions will be oscillating or not depending on whether a is positive (non-oscillating) or negative (oscillating) because To find the solutions for this choice of q(x), and verify the theorem for this example, substitute the 'Ansatz' which gives This means that (for non-zero a) the general solution is where A and B are arbitrary constants. It is not hard to see that for positive a 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, such as the Gesztesy–Ünal criterion.

Statement of the theorem due to H. Kneser

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: Let be a continuous function on the region, and such that for all. Given a real number c satisfying, define the set S_c as the set of points x_c for which there is a solution x = x(t) of such that x(t_0)=x_0 and x(c) = x_c. Then S_c is a closed and connected set.