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

${\displaystyle y''+q(x)y=0}$

with

${\displaystyle q:[0,+\infty )\to \mathbb {R} }$

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

${\displaystyle \limsup _{x\to +\infty }x^{2}q(x)<{\tfrac {1}{4}}}$

and oscillating if

${\displaystyle \liminf _{x\to +\infty }x^{2}q(x)>{\tfrac {1}{4}}.}$

Example

To illustrate the theorem consider

${\displaystyle q(x)=\left({\frac {1}{4}}-a\right)x^{-2}\quad {\text{for}}\quad x>0}$

where ${\displaystyle a}$ is real and non-zero. According to the theorem, solutions will be oscillating or not depending on whether ${\displaystyle a}$ is positive (non-oscillating) or negative (oscillating) because

${\displaystyle \limsup _{x\to +\infty }x^{2}q(x)=\liminf _{x\to +\infty }x^{2}q(x)={\frac {1}{4}}-a}$

To find the solutions for this choice of ${\displaystyle q(x)}$, and verify the theorem for this example, substitute the 'Ansatz'

${\displaystyle y(x)=x^{n}}$

which gives

${\displaystyle n(n-1)+{\frac {1}{4}}-a=\left(n-{\frac {1}{2}}\right)^{2}-a=0}$

This means that (for non-zero ${\displaystyle a}$) the general solution is

${\displaystyle y(x)=Ax^{{\frac {1}{2}}+{\sqrt {a}}}+Bx^{{\frac {1}{2}}-{\sqrt {a}}}}$

where ${\displaystyle A}$ and ${\displaystyle B}$ are arbitrary constants.

It is not hard to see that for positive ${\displaystyle a}$ the solutions do not oscillate while for negative ${\displaystyle a=-\omega ^{2}}$ the identity

${\displaystyle x^{{\frac {1}{2}}\pm i\omega }={\sqrt {x}}\ e^{\pm (i\omega )\ln {x}}={\sqrt {x}}\ (\cos {(\omega \ln x)}\pm i\sin {(\omega \ln x)})}$

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.[2]

References

1. Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
2. Helge Krüger and Gerald Teschl, Effective Prüfer angles and relative oscillation criteria, J. Diff. Eq. 245 (2008), 3823–3848