Kneser's theorem (differential equations)

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

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

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.

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 ${\displaystyle f\colon \mathbb {R} \times \mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}$ be a continuous function on the region ${\displaystyle {\mathcal {R}}:=[t_{0},t_{0}+a]\times \{x\in \mathbb {R} ^{n}:\Vert x-x_{0}\Vert \leq b\}}$, and such that ${\displaystyle |f(t,x)|\leq M}$ for all ${\displaystyle (t,x)\in {\mathcal {R}}}$.

Given a real number ${\displaystyle c}$ satisfying ${\displaystyle t_{0}<c\leq t_{0}+\min(a,b/M)}$, define the set ${\displaystyle S_{c}}$ as the set of points ${\displaystyle x_{c}}$ for which there is a solution ${\displaystyle x=x(t)}$ of ${\displaystyle {\dot {x}}=f(t,x)}$ such that ${\displaystyle x(t_{0})=x_{0}}$ and ${\displaystyle x(c)=x_{c}}$. Then ${\displaystyle S_{c}}$ is a closed and connected set.