Kneser's theorem (differential equations)

Kneser's theorem (differential equations) In Mathematik, auf dem Gebiet der gewöhnlichen Differentialgleichungen, the Kneser theorem, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not.

Inhalt 1 Aussage des Theorems 2 Beispiel 3 Erweiterungen 4 References Statement of the theorem Consider an ordinary linear homogeneous differential equation of the form {displaystyle y''+q(x)y=0} mit {Anzeigestil q:[0,+unendlich )zu mathbb {R} } kontinuierlich. 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 {Anzeigestil q(x)=links({frac {1}{4}}-recht)x^{-2}Quad {Text{zum}}quad x>0} wo {Anzeigestil a} is real and non-zero. Nach dem Satz, solutions will be oscillating or not depending on whether {Anzeigestil a} is positive (non-oscillating) or negative (oscillating) Weil {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 {Anzeigestil q(x)} , and verify the theorem for this example, substitute the 'Ansatz' {Anzeigestil y(x)=x^{n}} which gives {Anzeigestil n(n-1)+{frac {1}{4}}-a=left(n-{frac {1}{2}}Rechts)^{2}-a=0} Das bedeutet, dass (for non-zero {Anzeigestil a} ) the general solution is {Anzeigestil y(x)=Ax^{{frac {1}{2}}+{quadrat {a}}}+Bx^{{frac {1}{2}}-{quadrat {a}}}} wo {Anzeigestil A} und {Anzeigestil B} are arbitrary constants.

It is not hard to see that for positive {Anzeigestil a} the solutions do not oscillate while for negative {displaystyle a=-omega ^{2}} the identity {Anzeigestil x^{{frac {1}{2}}pm iomega }={quadrat {x}} e^{pm (iomega )ln {x}}={quadrat {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, Gerhard (2012). Gewöhnliche Differentialgleichungen und dynamische Systeme. Vorsehung: Amerikanische Mathematische Gesellschaft. ISBN 978-0-8218-8328-0. ^ Helge Krüger and Gerald Teschl, Effective Prüfer angles and relative oscillation criteria, J. Diff. Gl. 245 (2008), 3823–3848 [1] Kategorien: Ordinary differential equationsTheorems in analysisOscillation