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

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

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 ${\displaystyle f(t,x):R\times R^{n}\rightarrow R^{n}}$ be a continuous function on the region ${\displaystyle {\mathcal {R}}=[t_{0},t_{0}+a]\times \{x\in \mathbb {R} ^{n}:|x-x_{0}|\leq b\}}$, 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 ${\displaystyle x(c)=x_{c}}$. The set ${\displaystyle S_{c}}$ is a closed and connected set.