Kneser's theorem (differential equations)

Kneser's theorem (differential equations) En mathématiques, dans le domaine des équations différentielles ordinaires, the Kneser theorem, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not.

Contenu 1 Énoncé du théorème 2 Exemple 3 Rallonges 4 References Statement of the theorem Consider an ordinary linear homogeneous differential equation of the form {displaystyle y''+q(X)y=0} avec {style d'affichage q:[0,+infime )à mathbb {R} } continu. 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 {style d'affichage 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 {style d'affichage q(X)=gauche({frac {1}{4}}-un droit)x^{-2}quad {texte{pour}}quad x>0} où {style d'affichage a} is real and non-zero. D'après le théorème, solutions will be oscillating or not depending on whether {style d'affichage a} is positive (non-oscillating) or negative (oscillating) car {style d'affichage limsup _{xto +infty }x^{2}q(X)=liminf _{xto +infty }x^{2}q(X)={frac {1}{4}}-un} To find the solutions for this choice of {style d'affichage q(X)} , and verify the theorem for this example, substitute the 'Ansatz' {style d'affichage y(X)=x^{n}} qui donne {displaystyle n(n-1)+{frac {1}{4}}-a=left(n-{frac {1}{2}}droit)^{2}-a=0} Cela signifie que (for non-zero {style d'affichage a} ) the general solution is {style d'affichage y(X)=Ax^{{frac {1}{2}}+{sqrt {un}}}+Bx^{{frac {1}{2}}-{sqrt {un}}}} où {style d'affichage A} et {style d'affichage B} are arbitrary constants.

It is not hard to see that for positive {style d'affichage a} the solutions do not oscillate while for negative {displaystyle a=-omega ^{2}} the identity {style d'affichage x^{{frac {1}{2}}pm iomega }={sqrt {X}} e ^{pm (iomega )dans {X}}={sqrt {X}} (parce que {(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: Société mathématique américaine. ISBN 978-0-8218-8328-0. ^ Helge Krüger and Gerald Teschl, Effective Prüfer angles and relative oscillation criteria, J. Diff. Éq. 245 (2008), 3823–3848 [1] Catégories: Ordinary differential equationsTheorems in analysisOscillation