Kneser's theorem (differential equations)

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.

Contents 1 Statement of the theorem 2 Example 3 Extensions 4 References 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 _{xto +infty }x^{2}q(x)<{tfrac {1}{4}}} and oscillating if {displaystyle liminf _{xto +infty }x^{2}q(x)>{tfrac {1}{4}}.} Example To illustrate the theorem consider {displaystyle q(x)=left({frac {1}{4}}-aright)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 _{xto +infty }x^{2}q(x)=liminf _{xto +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 iomega }={sqrt {x}} e^{pm (iomega )ln {x}}={sqrt {x}} (cos {(omega ln x)}pm isin {(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 ^ Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. ^ Helge Krüger and Gerald Teschl, Effective Prüfer angles and relative oscillation criteria, J. Diff. Eq. 245 (2008), 3823–3848 [1] Categories: Ordinary differential equationsTheorems in analysisOscillation