Kneser's theorem (differential equations)

Kneser's theorem (differential equations) In matematica, nel campo delle equazioni differenziali ordinarie, the Kneser theorem, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not.

Contenuti 1 Enunciato del teorema 2 Esempio 3 Estensioni 4 References Statement of the theorem Consider an ordinary linear homogeneous differential equation of the form {displaystyle y''+q(X)y=0} insieme a {stile di visualizzazione q:[0,+infty )a matematicabb {R} } continuo. 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 {stile di visualizzazione q(X)= sinistra({frac {1}{4}}-giusto)x^{-2}quad {testo{per}}quad x>0} dove {stile di visualizzazione a} is real and non-zero. Secondo il teorema, solutions will be oscillating or not depending on whether {stile di visualizzazione a} is positive (non-oscillating) or negative (oscillating) perché {displaystyle 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 {stile di visualizzazione q(X)} , and verify the theorem for this example, substitute the 'Ansatz' {stile di visualizzazione y(X)=x^{n}} which gives {stile di visualizzazione n(n-1)+{frac {1}{4}}-a=left(n-{frac {1}{2}}Giusto)^{2}-a=0} Ciò significa che (for non-zero {stile di visualizzazione a} ) the general solution is {stile di visualizzazione y(X)=Ax^{{frac {1}{2}}+{mq {un}}}+Bx^{{frac {1}{2}}-{mq {un}}}} dove {stile di visualizzazione A} e {stile di visualizzazione B} are arbitrary constants.

It is not hard to see that for positive {stile di visualizzazione a} the solutions do not oscillate while for negative {displaystyle a=-omega ^{2}} the identity {stile di visualizzazione x^{{frac {1}{2}}pm iomega }={mq {X}} e^{pm (iomega )ln {X}}={mq {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. Provvidenza: Società matematica americana. 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] Categorie: Ordinary differential equationsTheorems in analysisOscillation